In my class we learned how to solve DEs using the variation of parameters when the coefficients are constant, we use undetermined coefficients to get two homogeneous solutions then apply the method of variation of parameters to get a particular solution. But I was never taught how to use method of variation of parameters when the coefficients are non-constant. For example
$$x^2y'' -3xy'+3y=12x^4$$
Has solutions $y_1=x$ and $y_2=x^3$.
I found this example online but even it doesn't show how these homogenous solutions were found… But it uses them to find the particular solution…. Which I'm fine with.
How do I find these solutions and is there a general strategy? Thanks!
Best Answer
Variation of parameters does not find the solutions of the homogeneous equation, it just uses them to find a particular solution of the non-homogeneous equation. Various other techniques can be used to find fundamental solutions of the homogeneous equation. For instance, this particular homogeneous equation $$ x^2 y'' - 3 x y' + 3 y = 0$$ is an Euler (or Cauchy-Euler, or Euler-Cauchy) equation, so you look for solutions of the form $y = x^r$, and find $x$ and $x^3$.
You can also use the method of reduction of order to find a second solution of the homogeneous equation, given one nontrivial solution.
But there is no completely general strategy to find closed-form solutions for homogeneous second-order linear differential equations with non-constant coefficients. Indeed, there are rather simple equations of this type that do not have nontrivial closed-form solutions, even if you allow the standard special functions. For example, as far as I know $$ y'' + y' + x^3 y = 0 $$ does not have them.