Linear Second-Order Differential Equations LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS
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1 11.11 LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS A Spring with Friction: Damped Oscillations The differential equation, which we used to describe the motion of a spring, disregards friction. But there is friction in every real system. For a mass on a spring, the frictional force from air resistance increases with the velocity of the mass. The frictional force is often approximately proportional to velocity, and so we introduce a damping term of the form, where a is a constant called the damping coefficient and is the velocity of the mass. Remember that without damping, the differential equation was obtained from With damping, the spring force is replaced by, where a is positive and the term is subtracted because the frictional force is in the direction opposite to the motion. The new differential equation is therefore which is equivalent to the following differential equation: Equation for Damped Oscillations of a Spring We expect the solution to this equation to die away with time, as friction brings the motion to a stop. The General Solution to a Linear Differential Equation The equation for damped oscillations is an example of a linear second-order differential equation with constant coefficients. This section gives an analytic method of solving the equation, for constant b and c. As we have seen for the spring equation, if and satisfy the differential Page 1 of 16
2 equation, then the principle of superposition says that, for any constants and, the function is also a solution. It can be shown that the general solution is of this form, provided. is not a multiple of Finding Solutions: The Characteristic Equation We now use complex numbers to solve the differential equation The method is a form of guess-and-check. We ask what kind of function might satisfy a differential equation in which the second derivative is a sum of multiples of and y. One possibility is an exponential function, so we try to find a solution of the form: where r may be a complex number. 22 To find r, we substitute into the differential equation: We can divide by provided, because the exponential function is never zero. If, then, which is not a very interesting solution (though it is a solution). So we assume. Then is a solution to the differential equation if This quadratic is called the characteristic equation of the differential equation. Its solutions are There are three different types of solutions to the differential equation, depending on whether the solutions to the characteristic equation are real and distinct, complex, or repeated. The sign of determines the type of solutions. The Case with There are two real solutions and to the characteristic equation, and the following two functions satisfy the differential equation: The sum of these two solutions is the general solution to the differential equation: Page 2 of 16
3 If, the general solution to is where and are the solutions to the characteristic equation. If and, the motion is called overdamped. A physical system satisfying a differential equation of this type is said to be overdamped because it occurs when there is a lot of friction in the system. For example, a spring moving in a thick fluid such as oil or molasses is overdamped: it will not oscillate. Example 1 A spring is placed in oil, where it satisfies the differential equation Solve this equation with the initial conditions and when. Solution The characteristic equation is with solutions and, so the general solution to the differential equation is We use the initial conditions to find and. At, we have Furthermore, since, we have Solving these equations simultaneously, we find and, so that the solution is The graph of this function is in Figure The mass is so slowed by the oil that it passes through the equilibrium point only once (when ) and for all practical purposes, it comes to rest after a short time. The motion has been damped out by the oil. Page 3 of 16
4 Figure 11.73: Solution to overdamped equation The Case with In this case, the characteristic equation has only one solution, both and are solutions.. By substitution, we can check that If, has general solution If, the system is said to be critically damped. The Case with In this case, the characteristic equation has complex roots. Using Euler's formula, 23 these complex roots lead to trigonometric functions which represent oscillations. Example 2 An object of mass is attached to a spring with spring constant, and the object experiences a frictional force proportional to the velocity, with constant of proportionality. Page 4 of 16
5 At time, the object is released from rest 2 meters above the equilibrium position. Write the differential equation that describes the motion. Solution The differential equation that describes the motion can be obtained from the following general expression for the damped motion of a spring: Substituting,, and, we obtain the differential equation: At, the object is at rest 2 meters above equilibrium, so the initial conditions are and, where s is in meters and t in seconds. Notice that this is the same differential equation as in Example 1 except that the coefficient of has decreased from 3 to 2, which means that the frictional force has been reduced. This time, the roots of the characteristic equation have imaginary parts which lead to oscillations. Example 3 Solve the differential equation subject to,. Solution The characteristic equation is The solution to the differential equation is where and are arbitrary complex numbers. The initial condition gives Also, Page 5 of 16
6 so gives Solving the simultaneous equations for and gives (after some algebra) The solution is therefore Using Euler's formula, and, we get Multiplying out and simplifying, all the complex terms drop out, giving The and terms tell us that the solution oscillates; the factor of tells us that the oscillations are damped. See Figure However, the period of the oscillations does not change as the amplitude decreases. This is why a spring-driven clock can keep accurate time even as it is running down. Figure 11.74: Solution to underdamped equation In Example 3, the coefficients and are complex, but the solution,, is real. (We expect this, since represents a real displacement.) In general, provided the coefficients b and c in the original differential equation and the initial values are real, the solution is always real. The coefficients and are always complex conjugates (that is, of the form ). If, to solve Page 6 of 16
7 Find the solutions to the characteristic equation. The general solution to the differential equation is, for some real and, If, such oscillations are called underdamped. If, the oscillations are undamped. Example 4 Find the general solution of the equations (a). (b).. Solution (a). The characteristic equation is, so. Thus the general solution is (b). The characteristic equation is, so. The general solution is Notice that we have seen the solution to this equation in Section 11.10; it's the equation of undamped simple harmonic motion. Example 5 Solve the initial value problem Page 7 of 16
8 Solution We solve the characteristic equation The general solution to the differential equation is Substituting gives Differentiating gives Substituting gives The solution is therefore Summary of Solutions to If If If, then, then, then Exercises and Problems for Section Click here to open Student Solutions Manual: Ch 11 Section 11 Click here to open Web Quiz Ch 11 Section 11 Exercises For Exercises 1 12, find the general solution to the given differential equation. Page 8 of 16
9 For Exercises 13 20, solve the initial value problem. 13.,,. 14.,,. 15.,,. 16.,,. 17.,, 18.,, 19.,, 20.,, For Exercises 21 24, solve the boundary value problem. Page 9 of 16
10 21.,,. 22.,,. 23.,, 24.,, Problems 25. Match the graphs of solutions in Figure with the differential equations below. (a). (b). (c). (d). Figure Match the differential equations to the solution graphs (I) (IV). Use each graph only once. (a). Page 10 of 16
11 (b). (c). (d). (I). (II). (III). (IV). Page 11 of 16
12 27. If is a solution to the differential equation find the value of the constant k and the general solution to this equation. 28. Assuming b,, explain how you know that the solutions of an underdamped differential equation must go to 0 as. For each of the differential equations in Problems 29 30, find the values of b that make the general solution: (a). overdamped, (b). underdamped, (c). critically damped Each of the differential equations (i) (iv) represents the position of a 1 gram mass oscillating on the end of a damped spring. For Problems 31 35, pick the differential equation representing the system which answers the question. (a). (b). (c). (d). 31. Which spring has the largest coefficient of damping? 32. Which spring exerts the smallest restoring force for a given displacement? Page 12 of 16
13 33. In which system does the mass experience the frictional force of smallest magnitude for a given velocity? 34. Which oscillation has the longest period? 35. Which spring is the stiffest? Hint: You need to determine what it means for a spring to be stiff. Think of an industrial strength spring and a slinky.] 36. Find a solution to the following equation which satisfies and does not tend to infinity as : 37. Consider an overdamped differential equation with b,. (a). Show that both roots of the characteristic equation are negative. (b). Show that any solution to the differential equation goes to 0 as. 38. Could the graph in Figure show the position of a mass oscillating at the end of an overdamped spring? Why or why not? Figure Consider the system of differential equations (a). Convert this system to a second order differential equation in y by differentiating the second equation Page 13 of 16
14 with respect to t and substituting for x from the first equation. (b). Solve the equation you obtained for y as a function of t; hence find x as a function of t. Recall the discussion of electric circuits. Just as a spring can have a damping force which affects its motion, so can a circuit. Problems involve a damping force caused by the resistor in Figure The charge Q on a capacitor in a circuit with inductance L, capacitance C, and resistance R, in ohms, satisfies the differential equation Figure If henry, ohms, and farads, find a formula for the charge when (a).,. (b).,. 41. If henry, ohm, and farads, find a formula for the charge when (a).,. (b).,. (c). How did reducing the resistance affect the charge? Compare with your solution to Problem 40. Page 14 of 16
15 42. If henry, ohm, and farads, find a formula for the charge when (a).,. (b)., (c). How did increasing the inductance affect the charge? Compare with your solution to Problem Given any positive values for R, L and C, what happens to the charge as t goes to infinity? 44. (a). If and satisfy show that is a solution to. (b). If, show that the solution in part (a) can be written (c). Using the Taylor series, show that (d). Use the result of part (c) in the solution from part (b) to show that. (e). If there is a double root. By direct substitution, show that satisfies. Page 15 of 16
16 Copyright 2006 John Wiley & Sons, Inc. All rights reserved. Page 16 of 16
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