Solve the following differential equations: `x(dy)/(dx)-y=(x-1)e^x`

Explanation

`frac{d y}{d x}-frac{y}{x}=frac{(x-1) e^{x}}{x} therefore P=frac{-1}{x}, Q=frac{(x-1) e^{x}}{x} `
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therefore int P d x=int frac{-1}{x} d x=-log x (text { take } log x>0) `
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therefore I F=e^{int P d x}=e^{-log x}=frac{1}{x}`


Solution is
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{y} times {IF}=int Q({IF}) {dx}+{c} ``
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Rightarrow y frac{1}{x}=int frac{(x-1) e^{x}}{x} frac{1}{x} d x+c ``
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