Please show full work Apply Variation of Parameters to find the general solution.e*у"- 2у'+ у%3D1+x?

Answered: Apply Variation of Parameters to find…

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter2: Working With Real Numbers

Section2.9: Dividing Real Numbers

Problem 9MRE

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Apply Variation of Parameters to find the general solution.
e*
у"- 2у'+ у%3D
1+x?

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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