On the theme of differential equations, I wonder why we still need to keep the solution of the homogeneous equation.
For example the linear differential equation :
$y' + ay = x^2 \enspace \enspace \enspace (E)$
$y' + ay = 0 \enspace \enspace \enspace (H)$
After resolving the homogeneous equation $(H)$, we get a first solution $y_h$ :
$y_h = k.e^{-ax}$
And after applying the Lagrange method, we resolve :
$ x^2 = k'(x) e^{-ax} \Longleftrightarrow k'(x)= x^2 e^{ax} $
$ k(x) = \int_{}^x t^2 e^{at} dt$
$k(x) = \Bigl(\frac{a^2x^2 - 2ax + 2}{a^3}\Bigr) e^{ax} $
So we get a particular solution $ y_0 $ as :
$ y_0 = k(x) e^{-ax} \Longleftrightarrow y_0 = \frac{a^2x^2 - 2ax + 2}{a^3} $
Now, we can verify that $ y_0 $ is conform with $(E)$ injecting $y_0$ and $y_0'$, which is the case.
So, why we have to say that the general solution is the sum of both solutions ($y_t = y_h + y_0$), whereas the solution $y_0$ works without adding $y_h$ ?
Here, i would be :
$y_t = y_h + y_0$
$y_t = k.e^{-ax} +\frac{a^2x^2 - 2ax + 2}{a^3} $
Thank you for your help.