Why do we find the homogeneous solution of inhomogeneous Differential Equations?
We all know that if we have inhomogeneous differential equation, we must solve for homogeneous solution and the inhomogeneous solution. And, in the end, we add them together for the complete solution.
Suppose i have simple ODE :
$y''+2y'+y=x$
And i got :
$y_h=\left(C_1+xC_2\right)e^{-x}\\ y_p=x-2\\$$ \begin{aligned} \therefore y &= y_h+y_p\\ &=\left(C_1+xC_2\right)e^{-x}+x-2 \end{aligned}$
But i know that the inhomogeneous solution ($y_p$) is satisfied enough for that ODE, so why don't we use the $y_p$ only?
Lately, This question appear in my mind and sometimes it's annoying cz i still can't answer it.
$\endgroup$ 31 Answer
$\begingroup$Note that in general we solve initial value problems. The reason for adding the homogenous solution to the particular solution is to find the general solution and be able to find solutions which satisfy any given initial conditions.
A particular solution satisfies a particular initial condition which is not necessarily what we are looking for.
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