Find a differential equation satisfied by all functions of some

Question

I am trying to work through a Problem set (beginning Masters Course) and I am completely stumped by this question.

Find a differential equation satisfied by all functions of some form

a) $f(x, y) = φ(x + y)$

b) $f(x, y) = φ(x) + ψ(y)$

c) $f(x, y) = χ [φ(x) + ψ(y)]$

Here $φ$, $ψ$, and $χ$ are arbitrary smooth functions of one variable.

I simply am not understanding what is even asked. I have learned about differential equations, but apparently not enough. I want to make it very clear that I am not necessarily just looking for answers here. I would just like to know what is asked and what a possible solution would look like and what steps might be taken to reach it. So if anyone would be willing to give me some guidance that would be great. Thank you very much in advance.

Answer 1

It should be a partial differential equation (PDE), not an ordinary differential equation (ODE). For example, for question a, it is:

$\dfrac{\partial f}{\partial x}=\dfrac{\partial f}{\partial y} \ $ this common value being equal to $\ \varphi'(x+y)$.

Answer 2

Well, the problem gives you a mission of creativity: figure out some possible partial differential equation that could give you these types of possible solution. Let's see, e.g, the first of the above possible functions

$$f(x,y) = \phi(x+y)$$

when he says that $\phi$ is smooth it means that this is a function that it has derivatives that are also continuous up to some order, in that case, we can put up to second order. Now, the first interesting thing in this functions is that it has a symmetry under the change of $x$ and $y$: $f(x,y) = \phi(x + y) = \phi(y+x) = f(y,x)$. This should tell you something about how is the partial differential equation this function should satisfies. I think this is the type of reasoning that this question is asking you to seek.

Look, for example, to the equation below:

$$\frac{\partial^2 f}{\partial x^2}(x,y) = \frac{\partial^2 f}{\partial y^2}(x,y)$$

You should think in that as a solution because of the properties $\partial (x+y)/\partial x = \partial (y+x)/\partial y$ (with the chain rule) or because of the symmetric solution that you have.

The second one for example, it is complete separable, so partial differential equations that are solvable by separable functions must be the one's to choose. For example a trivial one $\partial ^2 f/\partial x \partial y = 0$. I think that the question ask's more about the creativity to imagine (thinking in the relations that you now of calculus and the notions of what models partial differential equations) from where this functions came. This is the inverse problem in general.