The body encounters no air resistance. Find a the equation of motion in the coordinate system of Fig. Redo Problem 7. A ball of mass 5 slugs is dropped from a height of ft. A body of mass 2 kg is dropped from a height of m. Find the limiting velocity of the body if it encounters a resistance force equal to —50v.
A body of mass 10 slugs is dropped from a height of ft with no initial velocity. The body encounters an air resistance proportional to its velocity.
A body weighing 8 Ib is dropped from a great height with no initial velocity. As it falls, the body encounters a force due to air resistance proportional to its velocity. A body weighing Ib is dropped ft above ground with no initial velocity. A tank initially holds 10 gal of fresh water. Find a the amount and b the concentration of salt in the tank at any time t. Find the amount of salt in the tank when the tank contains exactly 40 gal of solution. A tank contains gal of brine made by dissolving 80 Ib of salt in water.
Find a the amount of salt in the tank at any time t and b the time required for half the salt to leave the tank. A tank contains gal of brine made by dissolving 60 Ib of salt in water. Find the amount of salt in the tank after 30 minutes. A tank contains 40 1 of solution containing 2 g of substance per liter.
Find the amount of substance in the tank after 15 minutes. A tank contains 40 1 of a chemical solution prepared by dissolving 80 g of a soluble substance in fresh water. Find the amount of substance in the tank after 20 minutes. Find a the transient current and b the steady-state current.
An RC circuit has an emf of volts, a resistance of 5 ohms, a capacitance of 0. Find a an expression for the charge on the capacitor at any time t and b the current in the circuit at any time t.
An RC circuit has no applied emf, a resistance of 10 ohms, a capacitance of 0. A RC circuit has an emf of 10 sin t volts, a resistance of ohms, a capacitance of 0. Find a the charge on the capacitor at any time t and b the steady-state current. Find a the charge on the capacitor at any time t and b the steady- state current. A RL circuit has an emf of 5 volts, a resistance of 50 ohms, an inductance of 1 henry, and no initial current. Find a the current in the circuit at any time t and b its steady-state component.
A RL circuit has no applied emf, a resistance of 50 ohms, an inductance of 2 henries, and an initial current of 10 amperes.
Find a the current in the circuit at any time t and b its transient component. A RL circuit has a resistance of 10 ohms, an inductance of 1. An RL circuit has an emf given in volts by 4 sin t, a resistance of ohms, an inductance of 4 henries, and no initial current.
Find the current at any time t. One hundred strands of bacteria are placed in a nutrient solution in which a plentiful supply of food is constantly provided but space is limited. The competition for space will force the bacteria population to stabilize at strands.
Under these conditions, the growth rate of bacteria is proportional to the product of the amount of bacteria present in the culture with the difference between the maximum population the solution can sustain and the current population. Estimate the amount of bacteria in the solution at any time t if it is known that there were strands of bacteria in the solution after seven hours. A new product is to be test marketed by giving it free to people in a city of one million inhabitants, which is assumed to remain constant for the period of the test.
It is further assumed that the rate of product adoption will be proportional to the number of people who have it with the number who do not. Estimate as a function of time the number of people who will adopt the product if it is known that people have adopted the product after four weeks. Find the velocity at any time t. In other words.
A linear differential equation has constant coefficients if all the coefficients bj x in 8. Theorem 8. Then 8. A set of functions is linearly dependent if there exists another set of constants, not all zero, that also satisfies 8.
If the only solution to 8. If the Wronskian is identically zero on this interval and if each of the functions is a solution to the same linear differential equation, then the set of functions is linearly dependent. Caution: Theorem 8. In this case, one must test directiy whether Eq.
Since none of these terms depends on y or any derivative of y, the differential equation is linear. None of these terms depends on y or any derivative of y; hence the differential equation is linear. See also Chapter 5. None of these terms depends on y or any of its derivatives, so the equation is linear. The equation is nonlinear because y is raised to a power higher than unity.
The equation is nonlinear because the first derivative of y is raised to a power other than unity, here the one-half power. Which of the linear differential equations given in Problem 8.
Using the results of Problem 8. Equations a , d , e , and h are nonhomogeneous linear differential equations. In their present forms, only c and h have constant coefficients, for only in these equations are all the coefficients constants.
The equation then becomes 8. Find the general form of a linear differential equation of a order two and b order one. Find the Wronskian of the set 8. Find the Wronskian of the set sin 3x, cos 3. This example shows that the Wronskian is in general a nonconstant function. The Wronskian of this set was found in Problem 8. Since it is nonzero for at least one point in the interval of interest in fact, it is nonzero at every point in the interval , it follows from Theorem 8. Redo Problem 8. Thus, the set is not linearly dependent; rather it is linearly independent.
In this case, Theorem 8. Consider the equation which can be rewritten as This linear equation can be satisfied for all x only if both coefficients are zero. Thus, the given set of functions is linearly dependent.
The Wronskian is identically zero and all functions in the set are solutions to the same linear differential equation, so it now follows from Theorem 8.
It follows, first from Theorem 8. It follows from Theorem 8. Two solutions of are e1 and Is the general solution We have It follows, first from Theorem 8. It was shown in Problems 8. Use the results of Problem 8. We have from Problem 8. It follows directly from Theorem 8. Consider the equation Recall that U3! The given set is, therefore, linearly independent. Do the results of Problems 8. Since the Wronskian of two linearly independent functions is identically zero, it follows from Theorem 8.
Does this result contradict the solution to Problem 8. Although W x3, U3! Does this result violate Theorem 8. Show that the second-order operator L y is linear; that is where c1 and c2 are arbitrary constants and y1 and y2 are arbitrary «-times differentiable functions.
In general, Thus 8. Prove Theorem 8. Supplementary Problems 8. Determine which of the linear differential equations in Problem 8. In Problems 8. Prove directly that the set given in Problem 8. Why doesn' t this result violate Theorem 8. Does Theorem 8.
Why doesn't this result violate Theorem 8. Chapter Equation 9. There are three cases to consider. Case 1. Case 2. Solved Problems 9. The characteristic equation is , which can be factored into Since the root!
Rewrite the solution of Problem 9. Using the results of Problem 9. Here the independent variable is t. Solve x - O. The characteristic equation is Using the quadratic formula, we find its roots to be These roots are a complex conjugate pair, so the general solution is given by 9. The characteristic equation is which can be factored into These roots are a complex conjugate pair, so the general solution is given by 9.
The characteristic equation is Using the quadratic formula, we find its roots to be These roots are a complex conjugate pair, so the general solution is 9. Solve The characteristic equation is Using the quadratic formula, we find its roots to be These roots are a complex conjugate pair, so the general solution is 9. The solution is given by 9. Prove that 9. Using Euler's relations we can rewrite 9. Since we are interested in the general real solution to 9.
Example The tharaL-terislii; i. If the roots A are all real and distinct, the solution is If the roots are distinct, but some are complex, then the solution is again given by As in Chapter 9, those terms involving complex exponentials can be combined to yield terms involving sines and cosines. These solutions are combined in the usual way with the solutions associated with the other roots to obtain the complete solution.
In theory it is always possible to factor the characteristic equation, but in practice this can be extremely difficult, especially for differential equations of high order. In such cases, one must often use numerical techniques to approximate the solutions. See Chapters 18, 19 and Solved Problems Compare this result with Problem 6.
Solve The characteristic equation, has roots and. The solution is If, using Euler's relations, we combine the first two terms and then similarly combine the last two terms, we can rewrite the solution as The solution is Find the general solution to a fourth-order linear homogeneous differential equation for y x with real numbers as coefficients if one solution is known to be J?
If x3e4x is a solution, then so too are X2e4x, xe4x, and e4x. We now have four linearly independent solutions to a fourth-order linear, homogeneous differential equation, so we can write the general solution as Determine the differential equation described in Problem The characteristic equation of a fourth-order differential equation is a fourth-degree polynomial having exactly four roots. If sin 3x is a solution, then so too is cos 3x.
The characteristic equation of a third-order differential equation must have three roots. Find the general solution to a sixth-order linear homogeneous differential equation for y x with real numbers as coefficients if one solution is known to be X2e7x cos 5. If x2elx cos 5x is a solution, then so too are xelx cos 5x and elx cos 5x. Furthermore, because complex roots of a characteristic equation come in conjugate pairs, every solution containing a cosine term is matched with another solution containing a sine term.
Consequently, x2elx sin 5x, xelx sin 5x, and elx sin 5x are also solutions. We now have six linearly independent solutions to a sixth-order linear, homogeneous differential equation, so we can write the general solution as Redo Problem An eighth-order linear differential equation possesses eight linearly independent solutions, and since we can only identify six of them, as we did in Problem We can say that the solution to Problem If xe2x is a solution, then so too is e2x which implies that X- 2 2 is a factor of the characteristic equation.
Determine the general solution of the differential equation. Determine the differential equation associated with the roots given in Problem Find the general solution to a fourth-order linear homogeneous differential equation for y x with real numbers as coefficients if two solutions are cos 4x and sin 3x. Find the general solution to a fourth-order linear homogeneous differential equation for y x with real numbers as coefficients if one solution is x cos 4x.
Find the general solution to a fourth-order linear homogeneous differential equation for y x with real numbers as coefficients if two solutions are xe2x and xe5x. A,, denote arhilrar; imiltiplicathe constants. Assume a solution of the form where Aj-. H is a constant to he determined. Assume a solution of the form where A is a constant to be determined. Case 3. Note: If f x is the sum or difference of terms already considered, then we take yp to be the sum or difference of the corresponding assumed solutions and algebraically combine arbitrary constants where possible.
From Problem 9. Using The general solution then is Again by Problem 9. Solve From Problem 9. From Problem Note, however, that this assumed solution has terms, disregarding multiplicative constants, in common with yh: in particular, the first-power term and the constant term. Using the results of Problem Then 1 becomes and the general solution is The solution also can be obtained simply by twice integrating both sides of the differential equation with respect to x.
Note, however, that this yp has exactly the same form as yh; therefore, we must modify yp. An assumed solution for x — 1 sin x is given by Eq. Then, from 1 , and the general solution is We are led, therefore, to the modified expression We now take yp to be the sum of 1 and 2 : Substituting 3 into the differential equation and simplifying, we obtain Equating coefficients of like terms, we have from which Equation 3 then gives and the general solution is Supplementary Problems In Problems In Problems Recall from Theorem 8.
This is permissible because we are seeking only one particular solution. This means that the system It is therefore more powerful than the method of undetermined coefficients, which is restricted to linear differential equations with constant coefficients and particular forms of j x.
Nonetheless, in those cases where both methods are applicable, the method of undetermined coefficients is usually the more efficient and, hence, preferable. As a practical matter, the integration of v' x may be impossible to perform. In such an event, other methods in particular, numerical techniques must be employed. This is a third-order equation with see Chapter 10 ; it follows from Eq.
Solve This is a third-order equation with see Chapter 10 ; it follows from Eq. Thus, Substituting these values into 1 , we obtain The general solution is therefore, This is a second-order equation for x t with It follows from Eq. The general solution is Solve In? We first write the differential equation in standard form, with unity as the coefficient of the highest derivative. The one exception is when ihc general solution is ihe homogeneous solution; lhai is, when the dilTerenlial equation under eon si derail on is iise.
The jicneral solution of the differential equation is given in Problem 1 I. I as Therefore. Solve The general solution of the differential equation is given in Problem Solve From Problem The solution to the initial-value problem is The system is in its equilibrium position when it is at rest. The mass is set in motion by one or more of the following means: displacing the mass from its equilibrium position, providing it with an initial velocity, or subjecting it to an external force F i.
A steel ball weighing Ib is suspended from a spring, whereupon the spring is stretched 2 ft from its natural length. The applied force responsible for the 2-ft displacement is the weight of the ball, Ib. For convenience, we choose the downward direction as the positive direction and take the origin to be the center of gravity of the mass in the equilibrium position.
We assume that the mass of the spring is negligible and can be neglected and that air resistance, when present, is proportional to the velocity of the mass.
Note that the restoring force Fs always acts in a direction that will tend to return the system to the equilibrium position: if the mass is below the equilibrium position, then x is positive and -kx is negative; whereas if the mass is above the equilibrium position, then x is negative and -kx is positive. The force of gravity does not explicitly appear in We automatically compensated for this force by measuring distance from the equilibrium position of the spring.
If one wishes to exhibit gravity explicitly, then distance must be measured from the bottom end of the natural length of the spring. The current 7 flowing through the circuit is measured in amperes and the charge q on the capacitor is measured in coulombs.
Kirchhojfs loop law: The algebraic sum of the voltage drops in a simple closed electric circuit is zero. It is known that the voltage drops across a resistor, a capacitor, and an inductor are respectively RI, HC q, and L dlldt where q is the charge on the capacitor. The voltage drop across an emf is —E t. Thus, from Kirchhoff s loop law, we have The relationship between q and 7 is Substituting these values into The second initial condition is obtained from Eq.
Thus, An expression for the current can be gotten either by solving Eq. See Problems Such a body experiences two forces, a downward force due to gravity and a counter force governed by: Archimedes' principle: A body in liquid experiences a buoyant upward force equal to the weight of the liquid displaced by that body.
Equilibrium occurs when the buoyant force of the displaced liquid equals the force of gravity on the body. Figure depicts the situation for a cylinder of radius r and height 77 where h units of cylinder height are submerged at equilibrium. At equilibrium, the volume of water displaced by the cylinder is 7tr2h, which provides a buoyant force of 7tr2hp that must equal the weight of the cylinder mg.
If the cylinder is raised out of the water by x t units, as shown in Fig. The downward or negative force on such a body remains mg but the buoyant or positive force is reduced to Jtr2[h - x t ]p. It now follows from Newton's second law that Substituting For buoyancy problems defined by Eq. For electrical circuit problems, the independent variable x is replaced either by q in Eq.
For damped motion, there are three separate cases to consider, depending on whether the roots of the associated characteristic equation see Chapter 9 are 1 real and distinct, 2 equal, or 3 complex conjugate. These cases are respectively classified as 1 overdamped, 2 critically damped, and 3 oscillatory damped or, in electrical problems, underdamped. A steady-state motion or current is one that is not transient and does not become unbounded.
Free undamped motion defined by Eq. Here c1, c2, and ft are constants with ft often referred to as circular frequency. The natural frequency j'is and it represents the number of complete oscillations per time unit undertaken by the solution. The period of the system of the time required to complete one oscillation is Equation The ball is started in motion with no initial velocity by displacing it 6 in above the equi- librium position.
The motion is free and undamped. Equation Find an expression for the motion of the mass, assuming no air resistance. The equation of motion is governed by Eq. Differentiating 2 , we obtain whereupon, and 2 simplifies to as the position of the mass at any time t. Determine the circular frequency, natural frequency, and period for the simple harmonic motion described in Problem Circular frequency: Natural frequency: Period: A kg mass is attached to a spring, stretching it 0.
The mass is started in motion from the equilibrium position with an initial velocity of 1 ml sec in the upward direction. Find the subsequent motion, if the force due to air resistance is i N.
Find the subsequent motion of the mass if the force due to air resistance is -2ilb. Find the subsequent motion of the mass, if the force due to air resistance is -lilb. Show that the types of motions that result from free damped problems are completely determined by the quantity a2 — 4 km.
The corresponding motions are, respectively, overdamped, critically damped, and oscillatory damped. Since the real parts of both roots are always negative, the resulting motion in all three cases is transient.
Find the subsequent motion of the mass if the force due to air resistance is iN. The equation of motion, These terms are the transient part of the solution. Assuming no air resistance, find the subsequent motion of the weight. This phenomenon is called pure resonance. It is due to the forcing function F t having the same circular frequency as that of the associated free undamped system. Write the steady-state motion found in Problem The steady-state displacement is given by 2 of Problem Substituting the given quantities into Eq.
Hence, As in Problem Solve Problem Substituting the values given in Problem Therefore, and as before. Note that although the current is completely transient, the charge on the capacitor is the sum of both transient and steady-state terms.
An RCL circuit connected in series has a resistance of 5 ohms, an inductance of 0. Find an expression for the current flowing through this circuit if the initial current and the initial charge on the capacitor are both zero. Substituting this value into 2 and simplifying, we obtain as before Determine the circular frequency, the natural frequency, and the period of the steady-state current found in Problem The current is given by 3 of Problem Write the steady-state current found in Problem Determine whether a cylinder of radius 4 in, height 10 in, and weight 15 Ib can float in a deep pool of water of weight density Let h denote the length in feet of the submerged portion of the cylinder at equilibrium.
Determine an expression for the motion of the cylinder described in Problem In the context of Fig. Determine whether a cylinder of diameter 10 cm, height 15 cm, and weight Let h denote the length in centimeters of the submerged portion of the cylinder at equilibrium. Let h denote the length of the submerged portion of the cylinder at equilibrium. A solid cylinder partially submerged in water having weight density Determine the diameter of the cylinder if it weighs 2 Ib.
We are given 0. The prism is set in motion by displacing it from its equilibrium position see Fig. Determine the differential equation governing the subsequent motion of this prism. For the prism depicted in Fig. By Archimedes' principle, this buoyant force at equilibrium must equal the weight of the prism mg; hence, We arbitrarily take the upward direction to be the positive x-direction.
If the prism is raised out of the water by x t units, as shown in Fig. It now follows from Newton's second law that Substituting 1 into this last equation, we simplify it to Fig. A lb weight is suspended from a spring and stretches it 2 in from its natural length. Find the spring constant. A mass of 0. It is then set into motion by stretching the spring 2 in from its equilibrium position and releasing the mass from rest.
Find the position of the weight at any time t if there is no external force and no air resistance. Find the position of the mass at any time t if there is no external force and no air resistance. A lb weight is attached to a spring, stretching it 8 ft from its natural length. Find the subsequent motion of the weight, if the medium offers negligible resistance.
Determine a the circular frequency, b the natural frequency, and c the period for the vibrations described in Problem Find the solution to Eq. A -slug mass is hung onto a spring, whereupon the spring is stretched 6 in from its natural length. Find the subsequent motion of the mass, if the force due to air resistance is —2x Ib.
A -j-slug mass is attached to a spring so that the spring is stretched 2 ft from its natural length. The mass is started in motion with no initial velocity by displacing it yft in the upward direction. Find the subsequent motion of the mass, if the medium offers a resistance of —4x Ib. The mass is set into motion by displacing it 6 in below its equilibrium position with no initial velocity. Find the subsequent motion of the mass, if the force due to the medium is —4x Ib.
Find the subsequent motion of the mass if the surrounding medium offers a resistance of -4iN. Find the subsequent motion of the mass, if the force due to air resistance is —4x Ib. A lb weight is attached to a spring whereupon the spring is stretched 1. Find the subsequent motion of the weight if the surrounding medium offers a negligible resistance.
A lb weight is attached to a spring whereupon the spring is stretched 2 ft and allowed to come to rest. Find the subsequent motion of the weight if the surrounding medium offers a resistance of —2x Ib. Write the steady-state portion of the motion found in Problem Find the subsequent motion of the mass if the surrounding medium offers a resistance of —3x N. Assuming no initial current and no initial charge on the capacitor, find expressions for the current flowing through the circuit and the charge on the capacitor at any time t.
Assuming no initial current and no initial charge on the capacitor, find an expression for the current flowing through the circuit at any time t. Determine the steady-state current in the circuit described in Problem An RCL circuit connected in series with a resistance of 16 ohms, a capacitor of 0.
Assuming no initial current and no initial charge on the capacitor, find an expression for the charge on the capacitor at any time t. Determine the steady-state charge on the capacitor in the circuit described in Problem Find the subsequent steady-state current in the circuit. Initial conditions are not needed. Find the steady-state current in the circuit. Hint Initial conditions are not needed. Determine the equilibrium position of a cylinder of radius 3 in, height 20 in, and weight 57rlb that is floating with its axis vertical in a deep pool of water of weight density Find an expression for the motion of the cylinder described in Problem Write the harmonic motion of the cylinder described in Problem Determine the equilibrium position of a cylinder of radius 2 ft, height 4 ft, and weight Ib that is floating with its axis vertical in a deep pool of water of weight density Determine the equilibrium position of a cylinder of radius 30 cm, height cm, and weight 2.
Find the general solution to Eq. Hint: Use the results of Problem The box is set into motion by displacing it x0 units from its equilibrium position and giving it an initial velocity of v0. Determine the differential equation governing the subsequent motion of the box.
Determine a the period of oscillations for the motion described in Problem II all the elements are numbers. Ihen the matrix is called a constant matrix. Matrices will prove to be very helpful in several ways. For example, we can recast higher-order differential equations into a sjslem of first-order differential equations using matrices see Chapter In particular, the first matrix is a constant matrix, whereas the last two are not.
A matrix is square if it has the same number of rows and columns. The third matrix given in Example I5. I is a vector. That is, Matrix addition is both associative and commutalue. Matrix multiplication is associative and distributes over addition; in general, however, it is not commutative. Theorem Cayley—Hamilton theorem. Any square matrix satisfies its own characteristic equation. That is, if then Solved Problems Find 3A - B for the matrices given in Problem Find 2A - B 2 for the matrices given in Problem But Therefore, the cancellation law is not valid for matrix multiplication.
Find Ax if Find Find J A dt for A as given in Problem Find the eigenvalues of A? Verify the Cayley-Hamilton theorem for the matrix of Problem Find a AB and b BA. Find A2. Find A7. FindB 2. Find a CD and b DC. Find a Ax and b xA. Find AC. Find the characteristic equation and eigenvalues of A. Find the characteristic equation and eigenvalues of B. Find the characteristic equation and eigenvalues of 3A.
Find the characteristic equation and the eigenvalues of C. Determine the multiplicity of each eigenvalue. Find the characteristic equation and the eigenvalues of D. Find for A as given in Problem Find Adt for A as given in Problem The infinite series However, it follows with some effort from Theorem I. Thus: Theorem Furthermore, if X; is an eigeinalue of multiplicity k.
When com- puting the various derivatives in Method of computation: For each eigenvalue A,, of A? When this is done for each eigenvalue, the set of all equations so obtained can be solved for a0, «i, These values are then substituted into Eq.
From Eq. Substituting these values successively into Eq. Substituting these values successively into It follows from Theorem From Eqs. It now follows from Theorem Now, according to Eq. Consider the following second- order differential equation: We see that!
The method of reduction is as follows. Step 1. Rewrite Thus, where and Step 2. Define n new variables the same number as the order of the original differential equation ; Xi t , x2 t , Express dxnldt in terms of the new variables. Proceed by first differentiating the last equation of Equations Define Then the initial conditions This last equa- tion is an immediate consequence of Eqs. The procedure is nearly identical to the method for reducing a single equation to matrix form; only Step 2 changes.
With a system of equations, Step 2 is generalized so that new variables are defined for each of the unknown functions in the set. Put the initial-value problem into the form of System Proceeding as in Problem Convert the differential equation into the matrix equation x? The given differential equation has no prescribed initial conditions, so Step 5 is omitted.
Put the following system into the form of System Put the following system into matrix form: We proceed exactly as in Problems Supplementary Problems Reduce each of the following systems to a first-order matrix system. In this chapter, and in the two succeeding chapters, we introduce several qualitative approaches in dealing with differential equations.
Observe that in a particular problem, f x , y may be independent of x, of y, or of x and y. A line element is a short line segment that begins at the point x, y and has a slope specified by A collection of line elements is a direction field.
The graphs of solutions to If the left side of Eq. When they are simple to draw, isoclines yield many line elements at once which is useful for constructing direction fields. To obtain a graphical approximation to the solution curve of Eqs. Denote the terminal point of this second line element as x2, y2.
Follow with a third line element constructed at x2, y2 and continue it a short distance. The process proceeds iteratively and concludes when enough of the solution curve has been drawn to meet the needs of those concerned with the problem. This formula is often written as where as required by Eq. In general, the smaller the step-size, the more accurate the approximate solution becomes at the price of more work to obtain that solution.
Thus, the final choice of h may be a compromise between accuracy and effort. If h is chosen too large, then the approximate solution may not resemble the real solution at all, a condition known as numerical instability. To avoid numerical instability, Euler's method is repeated, each time with a step-size one half its previous value, until two successive approximations are close enough to each other to satisfy the needs of the solver.
Continuing in this manner we generate the more complete direction field shown in Fig. To avoid confusion between line elements asso- ciated with the differential equation and axis markings, we deleted the axes in Fig. The origin is at the center of the graph. Describe the isoclines associated with the differential equation defined in Problem For the differential equation in Problem Some of these line elements are also drawn in Fig. Draw two solution curves to the differential equation given in Problem A direction field for this equation is given by Fig.
Two solution curves are shown in Fig. Observe that each solution curve follows the flow of the line elements in the direction field. Continuing in this manner, we generate Fig. At each point, we graph a short line segment emanating from the point at the specified angle from the horizontal. To avoid confusion between line elements associated with the differential equation and axis markings, we deleted the axes in Fig.
Draw three solution curves to the differential equation given in Problem Three solution curves are shown in Fig. These and other isoclines with some of their associated line elements are drawn in Fig. Draw four solution curves to the differential equation given in Problem Four solution curves are drawn in Fig.
Note that the differential equation is solved easily by direct integration. Observe that solution curves have different shapes depending on whether they are above both of these isoclines, between them, or below them. A representative solution curve of each type is drawn in Fig.
Give a geometric derivation of Euler's method. Give an analytic derivation of Euler's method. Let Y x represent the true solution. Then, using the definition of the derivative, we have If A. Thus, which is Euler's method. As before, Then, using Eq. Note that more accurate results are obtained when smaller values of h are used.
If we plot xn, yn for integer values of n between 0 and 10, inclusively, and then connect successive points with straight line segments, we would generate a graph almost indistinguishable from Fig. Find y 0. Then, using Eq. We proceed exactly as in Problem The results of these computations are given in Table The calculations are found in Table Supplementary Problems Direction fields are provided in Problems Sketch some of the solution curves.
See Fig. Describe the isoclines for the equation in Problem Find y 1. It is interesting to note that often the only required operations are addition, subtraction, multiplication, division and functional evaluations. In this chapter, we consider only first-order initial-value problems of the form Generalizations to higher-order problems are given in Chapter Remarks made in Chapter 18 on the step-size remain valid for all the numerical methods presented below.
The approximate solution at xn will be designated by y xn , or simply yn. The true solution at xn will be denoted by either Y xn or Yn. Note that once yn is known, Eq.
In general, the corrector depends on the predicted value. It then follows from Eq. The first of these values is given by the initial condition in Eq. The other three starting values are gotten by the Runge-Kutta method. In other words, if the true solution of an initial-value problem is a polynomial of degree n or less, then the approximate solution and the true solution will be identical for a method of order n.
In general, the higher the order, the more accurate the method. Euler's method, Eq. Then using Eqs. Compare it to Table From Then using Find y l.
Since the true solution is a second-degree polynomial and the modified Euler's method is a second-order method, this agreement is expected. Compare it with Table Thus, Then, using Eqs. Note that ys is significantly different from pys and y'4 is significantly different from py'4. When significant changes occur, they are often the result of numerical instability, which can be remedied by a smaller step-size.
Sometimes, however, significant differences arise because of a singularity in the solution. Figure is a direction field for this differential equation.
The cusp between 1. The analytic solution to the differential equation is given in Problem 4. The values of y0, yi, y2, y? Using Eqs. Then, using Eqs. In this chapter we investigate several numerical techniques dealing with such sjslems.
System In particular, The derivatives associated with the predicted values are obtained similarly, by replacing y and z in Eq. As in Chapter 19, four sets of starting values are required for the Adams-Bashforth-Moulton method.
The first set comes directly from the initial conditions; the other three sets are obtained from the Runge-Kutta method. We thus obtain the first-order system The given differential equation can be rewritten as or We thus obtain the first-order system Then, using Use the Runge-Kutta method to solve 3. It follows from Problem Use the Adams-Bashforth-Moulton method to solve 3. Formulate the Adams-Bashforth-Moulton method for System Formulate Milne's method for System All starting values and their derivatives are identical to those given in Problem Using the formulas given in Problem Obtain appropriate starting values from Table Formulate the modified Euler's method for System Approximation methods 6.
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Rigid Motion in … Expand. Mathematical Methods of Classical Mechanics. Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion.
Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid … Expand. Highly Influential.
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