Help: Solution to y+2x2y=x2y' + 2x^2y = x^2

Hi, I have been going through the courses for differential equations and just got done with the section on First Order DE. I decided to practice a few more problems on my own and I have been stumped for a while now because when I tried solving it I get different results and I cannot figure out what I am doing wrong. I think it's going to end up being such a small detail I missed but I cannot see it.

I was trying to solve it the way I saw here on Brilliant for a nonhomogeneous equation using the formula y(x)=A(x)ep(x)dxy(x) = A(x)e^{-\int p(x)dx} where A(x)=q(x)ep(x)dxA(x) = \int q(x)e^{\int p(x)dx} . When I solve for A(x)A(x) I get A(x)=x2e23x3dxA(x) = \int x^2e^{\frac{2}{3}x^3}dx then after u-substitution I get A(x)=12e23x3A(x) = \frac{1}{2}e^{\frac{2}{3}x^3}. Going back to the other formula y(x)=A(x)ep(x)dxy(x) = A(x)e^{-\int p(x)dx} then means y(x)=12e23x3e23x3y(x) = \frac{1}{2}e^{\frac{2}{3}x^3}e^{-\frac{2}{3}x^3} which is 12\frac{1}{2} which is not correct. The way the book does it is by seperation of variables but what am I doing wrong with using the above formulas?

Note by Christian Bracamontes
1 year ago

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