How do I continue? Which method should I relearn? $0.5u_{yy}+xu_y+x^2u=x^4$

Question

this question requires me to solve the PDE using methods from ODEs, unfortunately - I’ve managed to forgot the methods required to solve this.
what I did so far: $$0.5u_{yy}+xu_y+x^2u=x^4$$ $u=e^{ry}$ $$ 0.5r^2 +xr+x^2 =0 \implies r= -x\pm xi$$ $$ u_{h} = c_1(x) e^{-xy}\cos(xy) + c_2(x)e^{-xy}\sin(xy)$$ How do i find the particular? Any good sources you can recommend for relearning this are extremely appreciated. Thanks,

Answer 1

$$\frac12 u_{yy}+xu_y+x^2u=x^4$$ Note hat there is no $u_x$ in the equation. The only variable is $y$ with $x$ considered as a parameter. If this troubles you then change of symbol : $a\equiv x$. $$\boxed{\frac12 u''(y)+a\,u'(y)+a^2u(y)=a^4}$$ Solving the homogenous part $\frac12 u_h''(y)+a\,u_h'(y)+a^2u_h(y)=0$ leads to $$u_h=c_1e^{-ay}\sin(ay)+c_2e^{-ay}\cos(ay)$$ One have to add $u_p(y)$ a particular solution. In the present case $u_p(y)=a^2$ is obvious.

If no particular solution is obvious a method is the variation of coefficients $c_1$ and $c_2$ which become $f(y)$ and $g(y)$ using wronksian in the most general case :

https://mathworld.wolfram.com/VariationofParameters.html