Please show full work **Applying Variation of Parameters to Solve a Differential Equation**To find the general solution of the differential equation, use the method of Variation of Parameters:\[ y'' - 2y' + y = \frac{e^x}{1 + x^2} \]This method is used for solving non-homogeneous linear differential equations. The equation given is second-order and linear, making it suitable for this approach. The solution involves finding a particular solution to the non-homogeneous equation and adding it to the general solution of the corresponding homogeneous equation.

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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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**Applying Variation of Parameters to Solve a Differential Equation**

To find the general solution of the differential equation, use the method of Variation of Parameters:

\[ y'' - 2y' + y = \frac{e^x}{1 + x^2} \]

This method is used for solving non-homogeneous linear differential equations. The equation given is second-order and linear, making it suitable for this approach. The solution involves finding a particular solution to the non-homogeneous equation and adding it to the general solution of the corresponding homogeneous equation.

Expert Solution

Step 1

Given

$y'' - 2 y' + y = \frac{e^{x}}{1 + x^{2}}$

Step 2

Now

$y'' - 2 y' + y = \frac{e^{x}}{1 + x^{2}} T h e a u x i l i a r y e q u a t i o n i s m^{2} - 2 m + 1 = 0 m^{2} - m - m + 1 = 0 m (m - 1) - 1 (m - 1) = 0 (m - 1) (m - 1) = 0 m = 1, 1 T h e r e f o r e t h e c o m p l e m e n t a r y f u n c t i o n (C . F .) = A e^{x} + B x e^{x} . . . . . (1) N o w W (y_{1}, y_{2}) = |\begin{matrix} y_{1} & y_{2} \\ y_{1}' & y_{2}' \end{matrix}| F r o m e q u a t i o n (1), y_{1} = e^{x}, y_{2} = x e^{x} a n d y_{1}' = e^{x}, y_{2}' = (x + 1) e^{x} T h e n W (y_{1}, y_{2}) = |\begin{matrix} e^{x} & x e^{x} \\ e^{x} & (x + 1) e^{x} \end{matrix}| W (y_{1}, y_{2}) = (x + 1) e^{2 x} - x e^{2 x} W (y_{1}, y_{2}) = e^{2 x}$

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