the form of the solution of a linear ODE

Hi everyone.

I'm having a little trouble understanding something, so maybe (hopefully) you guys can help.

It is a well known fact in differential equations that the solution to a first order linear differential equation is the sum of the solution to the associated homogeneous equation and the solution to the particular equation, that is

$\displaystyle y=y_c+y_p$, where $\displaystyle y_c$ is the solution of

$\displaystyle \frac{dy}{dx}+P(x)y=0$

and $\displaystyle y_p$ of

$\displaystyle \frac{dy}{dx}+P(x)y=f(x) $

Now, I understand that this is a truism because

$\displaystyle \frac{d}{dx}[y_c+y_p]+P(x)[y_c+y_p]=\underbrace{\frac{dy_c}{dx}+P(x)y_c}_0+\underbrac e{\frac{dy_p}{dx}+P(x)y_p}_{f(x)}=f(x)$,

but what I'm having trouble with is that *of course *if I add zero to anything it won't change the element at all because of the fact that we are working on the real numbers here. So, what is the purpose here? I mean, I could easily say that...

The number 5 has the property that it is the sum

$\displaystyle \frac{dy_c}{dx}+P(x)y_c+5=5$.

But to what end?

By the way. I'm really sorry if this question makes no sense to you. If it doesn't, just tell me and I'll try to figure out what my malfunction is on my own.

Thank you all for reading.