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

Explanation

Given, `x frac{d y}{d x}-y=(x-1) e^{x}`


dividing both sides by `x`, we get


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`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}`


`therefore int P d x=int frac{-1}{x} d x=-log x quad(text { take } quad log x>0)`


`therefore I F=e^{int P d x}=e^{-log x}=frac{1}{x}`


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