When do I have to respect the $C$ constant and when can I combine?

Question

Question

Verify that the given two-parameter family of functions is the general solution of the non-homogeneous differential equation on the indicated interval.

$$ y''-4y'+4y = 2e^{2x}+4x-12 $$ $$ y=C_1e^{2x}+C_2xe^{2x}+x^2e^{2x}+x-2; (-\infty,\infty) $$

I solved it to the point where, after plugging in my $y''$ I get this on the left side of the top equation:

$$ -4C_1e^{2x}-4C_2xe^{2x}-4x^2e^{2x}+2e^{2x}-4 $$

I'm not 100% positive if the above is correct but my curiosity compulsed me to come here and ask this question: Could I combine the $C_1$ and $C_2$ if they happened to have the same $x$ terms? For example, let's say they read $-4C_1xe^{2x}$ and $-4C_2xe^{2x}$. Would this be equal to $-8Cxe^{2x}$? When do I have to respect the difference in $C$'s and when not?

Answer 1

The general solution of a linear ODE can be written as $$ y=y_{h}+y{c} $$ where $y_{h}$ stands for the homogeneous solution and $y_{c}$ is the complementary particular solution. In your case, the ODE $$ y''-4y'+4y=0 $$ has homogeneous solutions $c_{1}e^{2x}$ and $c_{2}xe^{2x}$.

You can easily verify that $x^2e^{2x}+x-2$ is particular solution for $$ y''-4y'+4y = 2e^{2x}+4x-12 $$ With that you are done. $c_{1}$ and $c_{2}$ will be determined from the appropriate initial conditions. I hope that helps.

Answer 2

What you are suggesting is fine. In general, you can always define $C_3=f(C_1,C_2)$ as long as doing so eliminates either $C_1$ or $C_2$ from the equation. While it is still technically correct otherwise, you wind up with three constants which are then no longer arbitrary, which only serves to make the solution more confusing.