Find the general solution of the differential equation y-y-et using (a) a method (without Laplace transform)...

 

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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. Find the solution of the initial value problem y² +4y=g(t) = { 16 0<t<7, y(0) 0, '(0) 0 using (a) a method (without Laplace transform) given in Chapter 3 or 4, (b) the Laplace transform. 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)= B, where You may first solve the IVP g(t)= { gi(t), 0<t<# 92(t), t2. y"+by+cy gr(t); y(0) a, y(0) - B on interval 0 < t < to obtain the solution, say, yi(t). Then you can solve the IVP y" +by+cy=92(t); y) = (1), ✓ (1) - « (1) y₁ on interval t > to obtain the solution, say, y2(t). (You are able to evaluate yı (4) and y(), the initial values at t = 4, since you have obtained y(t) already.) The solution of the original IVP then is 3(t) = { 32(t), 0<t< t> 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. Find the solution of the initial value problem y² +4y=g(t) = { 16 0<t<7, y(0) 0, '(0) 0 using (a) a method (without Laplace transform) given in Chapter 3 or 4, (b) the Laplace transform. 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)= B, where You may first solve the IVP g(t)= { gi(t), 0<t<# 92(t), t2. y"+by+cy gr(t); y(0) a, y(0) - B on interval 0 < t < to obtain the solution, say, yi(t). Then you can solve the IVP y" +by+cy=92(t); y) = (1), ✓ (1) - « (1) y₁ on interval t > to obtain the solution, say, y2(t). (You are able to evaluate yı (4) and y(), the initial values at t = 4, since you have obtained y(t) already.) The solution of the original IVP then is 3(t) = { 32(t), 0<t< t>

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a Method without Laplace transform Given the differentia... View the full answer