Integrating Factor for First Order Linear Differential Equation

[wowmaths]

Homework Statement

Find an integrating factor for the first order linear differential equation
$\frac{dy}{dx} - \frac{y}{x} = xe^{2x}$
and hence find its general solution

Homework Equations

The Attempt at a Solution

I found the integrating factor which is $e^{-lnx} = x^{-1}$

and multiplying the equation with the integrating factor, will result in:
$\frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = e^{2x}$

how do I go on from here?

 

[tiny-tim]

hi wowmaths!

the LHS is the exact derivative of … ? ​

 

[HallsofIvy]

Do you know why you found the "integrating factor"?

The whole point of an integrating factor for $dy/dx+ a(x)y= f(x)$ is that, with integrating factor $\mu(x)$, we will have
$$\mu(x)\frac{dy}{dx}+ \mu(x)a(x)y= \frac{d(\mu(x)y}{dx}= \mu(x)f(x)$$

If $\mu(x)= 1/x$ here (I have not checked that) then your equation should reduce to
$$\frac{d(y/x)}{dx}= e^{2x}$$
Integrate both sides of that with respect to x.