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#### karush

##### Well-known member

- Jan 31, 2012

- 2,928

$$\displaystyle y^\prime +y = xe^{-x}+1$$

$\textit{Solve the given differential equation}$

$\textit{From:$\displaystyle\frac{dy}{dx}+Py=Q$}$

$\textit{then:}$

$$\displaystyle

e^x y=\int x+e^{-2x} \, dx

+ c \\

\displaystyle e^x y=\frac{1}{2}(x^2-e^{-2x})+c$$

$\textit{divide every term by $e^x$}$

$$\displaystyle y=\frac{1}{2(e^x)}(x^2)

-\frac{e^{-2x}}{2(e^x)}+\frac{c}{(e^x)}$$

$\textit{simplify and reorder terms}$

$$\displaystyle y=c_1e^{-x}+\frac{1}{2}e^{-x}x^2+1$$

$\textit{Answer by W|A}$

$$y(x)=\color{red}

{\displaystyle c_1e^{-x}+\frac{1}{2}e^{-x}x^2+1}$$

any bugs any suggest???