Question

Find the general solution of the equation: x dy/dx + y = xe^x

Answer 1

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.