Why do we get particular solution when solving non homogeneous differential equation and not general solution

Question

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?

Answer 1

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.)

Answer 2

What you are observing here is a fundamental principle valid in the "linear world". You are given an equation or system of equations $$Ax=b\ ,\tag{1}$$ whereby $A$ operates linearly on the input vector $x$, and $b$ is a given constant vector. Such an equation may have no solutions. If it has solutions then they are of the form $$x=y_{\rm hom} +x_p\ ,$$ whereby $x_p$ is a particular solution of the original equation $(1)$ (maybe found by guessing), and $y_{\rm hom}$ is the general solution of the associated homogeneous equation $$Ay=0\ .\tag{2}$$ Note that the set of solutions of $(2)$ is a vector space, which means that any linear combination of "special solutions" of $(2)$ found by "special methods" is again a solution of $(2)$. One last thing: If the RHS of $(1)$ is the sum of two "simpler" vectors: $b=b_1+b_2$, and we can find "particular solutions" $x_p^{(1)}$, $x_p^{(2)}$ of $(1)$ for these "simpler" RHSs then $x_p=x_p^{(1)}+x_p^{(2)}$ is a solution of $(1)$ for the given $b$.

What I have described here applies to inhomogeneous linear systems of ordinary equations, to inhomogeneous linear systems of ODEs, and to inhomogeneous linear PDEs with homogeneous and inhomogeneous boundary conditions. It is a truly fundamental principle: The set of all solutions of $(1)$ is an affine space, i.e., a linear space of functions $V$, albeit translated away from the origin by some (not uniquely defined) vector $x_p$.