0
$\begingroup$

Let's say we're given the ode: $$\frac{d^2y}{dx^2}+\frac{dy}{dx}-2y=0$$

The general solution to this ode is given by: $$y(x)=c_1e^x+c_2e^{-2x}$$ where $c_1$ and $c_2$ are constants.

Now, we if multiply this general solution by a constant $b$, then this new equation also satisfies the ode, i.e. $$y(x)=bc_1e^x+bc_2e^{-2x}$$ is also a solution to the original ode.

Questions:

  1. Does this only apply to all linear homogeneous ode's?
  2. What is the name of this particular rule? (I think it's just a general linearity property, but aren't sure - correct me if I'm wrong)
  3. This applies to both ode's and pde's?
$\endgroup$
3
  • $\begingroup$ You sure multiplying by $b$ still gives a solution in your example? I think it holds for DEs homogeneous in $y$ only. $\endgroup$ – velut luna Oct 23 '17 at 0:24
  • $\begingroup$ @velutluna Oh, yes. I just checked and you're right. But my question still holds, in a slightly different light - what is the name of this rule (which applied to homogeneous equations)? $\endgroup$ – YinWai Tse Oct 23 '17 at 0:27
  • $\begingroup$ See the part homogeneous linear DE: en.wikipedia.org/wiki/Homogeneous_differential_equation $\endgroup$ – velut luna Oct 23 '17 at 0:36
1
$\begingroup$

Only the complementary function can contain arbitrary constants. The particular integral is exactly that: particular to the given right-hand side, and cannot be changed.

The analogue is that the general solution of a matrix equation $Ax=b$, for an $A$ with a nontrivial kernel (that is, the set of vectors it maps to zero: $\{y:Ay=0\}$) may be written as $x = y + c$, where $c$ is some particular vector so that $Ac=b$ (analogous to the particular integral), and $y$ is an arbitrary vector in the kernel (as the complementary function is for the homogeneous operator equation $Lu = 0$, where $L$ is a differential operator).

This is strictly linear phenomenon. The solution space in both cases is an affine space for inhomogeneous equations (akin to a plane not through the origin), but a linear (i.e. vector) space for homogeneous ones. Hence solutions can be added and multiplied by constants for the latter, but not the former. The whole point of the operator being linear is that $L(au_1+bu_2) = aLu_1+bLu_2$. I don't think this has a specific name, probably due to its fundamental nature. [EDIT: of course it's the principle of superposition] A more advanced, but related phenomenon is the Fredholm alternative.

$\endgroup$
0
$\begingroup$
  1. This also apply to some nonlinear equations as $y'=\frac{y^2}{y'+y}$.

  2. The concept of linearity includes two notions but you only refers to the homogeneity. So, in the context of the linear differential equations, this particular rule can be seen as a particular case of the principle of superposition.

  3. Yes.

$\endgroup$

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.