Integrating Factor for First Order Linear Differential Equation

wowmaths
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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?
 
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hi wowmaths! :smile:

the LHS is the exact derivative of … ? :wink:
 
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.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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