Differential Equations with xe^(2x) on the right side

[ASZ]

I'm not exactly sure how to solve a problem like this. My first thought was to use y=Ax+Be^(2x) and then y' would be y'=A+2Be^2x but that seems to be giving me a very strange result.

Lets say the problem was y'-6y=xe^2x

How would you determine the y= solution to use in solving this?

 

[galactus]

This is not as involved as undetermined coefficients.

Use the integrating factor

\(\displaystyle e^{\int -6dx}=e^{-6x}\)

\(\displaystyle \frac{d}{dx}[ye^{-6x}]=xe^{2x}e^{-6x}\)

Now, integrate both sides and solve for y.

Have you seen this method before?.

 

[ASZ]

so I'd get the anti-derivative 1/2y²-y²=1/3xe^2x?
but then how would I solve for y when neither x or y are provided

btw the actual problem is y'-2y=xe^2x which is why I got the answer I mentioned above

 

[galactus]

Do the same as I done above except use -2 instead of -6.

Remember, when you integrate a derivative, the function remains.

\(\displaystyle \displaystyle \int\frac{d}{dx}[ye^{-2x}]=ye^{-2x}\)

Don't forget the constant of integration on the right side.

\(\displaystyle \int xe^{2x}\cdot e^{-2x}=\int xdx=\frac{x^{2}}{2}+C\)

So, we have \(\displaystyle ye^{-2x}=\frac{x^{2}}{2}+C\)

Now, solve for y.

\(\displaystyle y=\left(\frac{x^{2}}{2}+C\right)e^{2x}\)

The integrating factor of \(\displaystyle \displaystyle\frac{dy}{dx}+p(x)y=g(x)\) is

\(\displaystyle e^{\int p(x)dx}\)

You appear to be familiar with undetermined coefficients yet unfamiliar with solving a linear DE using an integrating factor?.

The latter is dealt with earlier in a typical DE course. As a matter of fact, the integrating factor is often brought up in calc II.

 

[ASZ]

there's a few things in that demonstration that I dont seem to understand. where did the y' go?
does the -2 in -2y disappear when you raise e to the power of that number? and is the "x" always there whenever it is raised because of the integrating factor formula?

and the other part which confuses me is how you got x²/2 from multiplying xe^2x with e^-2x