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1. Find the general solution of the differential equation y"-y=c²t using (a) a method (without Laplace transform) given in Chapter 3 or 4; (b) the Laplace transform. Hint: In order to use the Laplace transform, you need to use the initial conditions as y(0) = a and y'(0) = b, where a, b are real (arbitrary) constants.

1. Find the general solution of the differential equation y"-y=c²t using (a) a method (without Laplace transform) given in Chapter 3 or 4; (b) the Laplace transform. Hint: In order to use the Laplace transform, you need to use the initial conditions as y(0) = a and y'(0) = b, where a, b are real (arbitrary) constants.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1. Find the general solution of the differential equation y" - y = e²t using (a) a method (without Laplace transform) given in Chapter 3 or 4; (b) the Laplace transform. Hint: In order to use the Laplace transform, you need to use the initial conditions as y(0) = a and y'(0) = b, where a, b are real (arbitrary) constants. 2. Find the solution of the initial value problem y" + 4y = g(t) = { 1° 1 > 16 0<t< where 7 using (a) a method (without Laplace transform) given in Chapter 3 or 4, (b) the Laplace transform. You may first solve the IVP Hint: The methods in Chapter 3 or 4 can be used to solve the nonhomogeneous equation with piecewise continuous right-hand side function. Here is the basic idea. Consider the initial value problem: y"+ay+by=g(t); y(0) = a, y'(0) = ß, y(0) 0, '(0) = 0 92(t), t>. y" + by' + cy= gi(t); y(0) = a, y'(0) = ß on interval 0 < t < to obtain the solution, say, y(t). Then you can solve the IVP y" +by+cy=92(t); ¹ (7)=»(7), «(-7)= « (7) on interval t > to obtain the solution, say, y2(t). (You are able to evaluate y₁ (4) and y(), the initial values at t = 1, since you have obtained y₁ (t) already.) The solution of the original IVP then is y(t) = { vh(t), 0<t< t>
Transcribed Image Text:1. Find the general solution of the differential equation y" - y = e²t using (a) a method (without Laplace transform) given in Chapter 3 or 4; (b) the Laplace transform. Hint: In order to use the Laplace transform, you need to use the initial conditions as y(0) = a and y'(0) = b, where a, b are real (arbitrary) constants. 2. Find the solution of the initial value problem y" + 4y = g(t) = { 1° 1 > 16 0<t< where 7 using (a) a method (without Laplace transform) given in Chapter 3 or 4, (b) the Laplace transform. You may first solve the IVP Hint: The methods in Chapter 3 or 4 can be used to solve the nonhomogeneous equation with piecewise continuous right-hand side function. Here is the basic idea. Consider the initial value problem: y"+ay+by=g(t); y(0) = a, y'(0) = ß, y(0) 0, '(0) = 0 92(t), t>. y" + by' + cy= gi(t); y(0) = a, y'(0) = ß on interval 0 < t < to obtain the solution, say, y(t). Then you can solve the IVP y" +by+cy=92(t); ¹ (7)=»(7), «(-7)= « (7) on interval t > to obtain the solution, say, y2(t). (You are able to evaluate y₁ (4) and y(), the initial values at t = 1, since you have obtained y₁ (t) already.) The solution of the original IVP then is y(t) = { vh(t), 0<t< t>
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