Question

solve the initial value problem:
dy/dx - y/x = xe^x , y(1) = e - 1

I know the answer is y = xe^x - x i just dont know the steps to get to it

Answer 1

The integrating factor is 1/x

So (1/x)y' + (-1/x^2)y = e^x
i.e.,
[ (1/x)y ]' = e^x
thus
(1/x)y = e^x + c

=> y = x.e^x + cx

Now apply your initial condition.