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

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter3: Functions And Graphs

Section3.3: Lines

Problem 26E

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Concept explainers

Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.

Question

Please show full work

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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