Find the general solution of the equation:

x dy/dx + y = xe^x

Step-by-step explanation:

x * (dy/dx) + y = x * (dy/dx) + 1*y=x * (dx/dy) + dx/dx*y = (d (xy) / dx)

d (xy) / dx = x^ex

d (xy) = xe^x*dx

We integrate both parts with their respective variables.

Integral of d (xy) is xy+c.

Integral of xe^x*dx is equal to xe^x - e^x+c.

We have to integrate by parts:

Integral (x*e^x) = x*e^x - integral ((x) 'e^x) = x*e^x - integral (e^x) = x*e^x-e^x.

We have that xy + c = x*e^x - e^x + c1, where c and c1 are non defined real constants.

So we get that xy = x*e^x - e^x + c2, where c2 is a real constant.