1
$\begingroup$

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?

$\endgroup$
4
  • $\begingroup$ Because there would be no way to find it without using the homogeneous equation first. Any particular solution of a linear inhomogeneous ODE added to the general solution of its homogeneous part gives the general solution to the original equation. It's how it works $\endgroup$
    – Yuriy S
    Mar 10, 2018 at 18:37
  • $\begingroup$ In principle we can try to directly solve the inhomogeneous ODE, for example by a series method, but it would need to be done from scratch for each equation $\endgroup$
    – Yuriy S
    Mar 10, 2018 at 18:38
  • $\begingroup$ Note that you should mention in the question that this method only works for linear ODEs $\endgroup$
    – Yuriy S
    Mar 10, 2018 at 18:39
  • 1
    $\begingroup$ Why do you want this? If you have two linear ODE's that have the same homogeneous part and different forcing functions, isn't it better to be able to reuse the solution of the homogeneous equation? $\endgroup$
    – saulspatz
    Mar 10, 2018 at 18:57

2 Answers 2

4
$\begingroup$

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.)

$\endgroup$
7
  • $\begingroup$ I would add to this that from the practical standpoint, knowing the homogeneous solution, we can use it to find a particular solution for the inhomogeneous equation as well. Which is generally better than trying to guess $\endgroup$
    – Yuriy S
    Mar 10, 2018 at 19:11
  • $\begingroup$ Sure, and agreed (e.g. variation of parameters), but that's not what the OP asked. The general solution of any linear equation arises from translates of the kernel of the homogeneous equation. $\endgroup$ Mar 10, 2018 at 19:14
  • $\begingroup$ @symplectomorphic I still don't understand why it is so that the solution is particular solution + general solution of homo. Is it some fundamental nature of diff equations that we stumbled upon? $\endgroup$
    – Allen
    Mar 11, 2018 at 7:22
  • $\begingroup$ Also what I am asking is that why is it that we get a particular solution and not a general solution when we solve? $\endgroup$
    – Allen
    Mar 11, 2018 at 7:34
  • $\begingroup$ It is not the fundamental nature of differential equations, but rather the fundamental nature of linear equations, whether differential or not. The idea doesn't work for nonlinear differential equations. The answer to your second question depends on what you mean by "when we solve." There are a variety of methods for finding one solution to an inhomogeneous equation, but as my answer shows, there are really infinitely many answers: once you find one particular solution, you get others by adding a homogeneous solution. $\endgroup$ Mar 11, 2018 at 16:14
1
$\begingroup$

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 $x_p$.

$\endgroup$
2
  • $\begingroup$ So what you are saying is that this is a fundamental nature of every linear equation that difference of two solution is a solution of homogeneous equation. $\endgroup$
    – Allen
    Mar 11, 2018 at 7:25
  • $\begingroup$ Also what I am asking is that why is it that we get a particular solution and not a general solution when we solve? $\endgroup$
    – Allen
    Mar 11, 2018 at 7:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.