What I don't understand is that why can't we find the general solution of non homogeneous differential equation from the non homogeneous one itself. Currently we use the homogeneous equation also.
Why isn't it that general solution is not available from the non-homogeneous equation itself?
Best Answer
The whole point is that if $x_1$ and $x_2$ solve $L(x)=y$, where $L$ is a linear differential operator, then $$0=y-y=L(x_1)-L(x_2)=L(x_1-x_2)$$ so $x_1$ and $x_2$ differ by a homogeneous solution. Note that linearity is crucial here.
(You seem to think the inhomogeneous equation has a unique particular solution. But this is not true. When you solve an inhomogeneous equation, you find a particular solution of the infinitely many that are available. The other solutions to the equation differ from the one you find by a homogeneous solution, that is, by an element of the kernel of the linear operator.)