A question about homogeneous ODE's and the usefulness of certain solutions

Question

If in an attempt to solve an ODE one happens arrive at a point where one deals with an equation of the type $g(x,y)=h(x,y)$ and one is unable to solve for $y$ and express it in terms of some function $f(x)$, are there any circumstances under which one still is able to attain useful information about a particular problem involving the ODE or is the result essentially useless? I understand that this is a rather vague question but let me explain why I'm asking this. I have book which serves as an introduction to ODE theory and in the chapter of first order equations, homogeneous equations are covered, you know the ones that involve some function with the $f(tx,ty)=f(x,y)$ property and then one sets $z=y/x$ in an attempt to solve the equation. However, the solutions that I attain while solving the assignments do not allow me to ultimately solve for $y$. I get to the point where my equations are of the type I mentioned in the beginning. The assignments are deliberately constructed for this to be the case, but does that in some sense count as solving the equation? Can I in some context acquire useful information without acutally being able to solve for $y$ just by getting to that point?

Answer 1

The question about homogeneous ODE's and the usefulness of certain solutions isn't specific of ODE's, but raises the question of usefulness of equations in general.

What means usefulness ? Is an implicit equation more ore less useful than a Cartesian equation or a polar equation, or other equivalent representation in different systems ?

No definitive answer can be proposed : This depends on cases, circumstances and context. The example that you gave is an excellent illustration. The general solution of the ODE $$\frac{dy}{dx}=\frac{x+y}{x-y}$$ is expressed on the form of implicit equation : $$\tan^{-1}\left(\frac{y}{x}\right)-\frac{1}{2}\ln(x^2+y^2)=c$$ This is in Cartesian system of coordinates. This form of solution seems of few usefulness at first sight because an explicit function $y=f(x)$ cannot be derived. But why looking for solving the implicit equation for $y$ ? Nothing proves that it is the better way to acquire useful information. Obviously $\frac{y}{x}$ and $\ln(x^2+y^2)$ suggest to express the solution in polar coordinates : $\begin{cases}x=\rho\cos(\theta) \\ y=\rho\sin(\theta) \end{cases}$ $$\theta-\ln(\rho)=c$$ So, the implicit equation becomes explicit : $$\rho=e^{\theta-c}=C\:e^{\theta}$$ We identify a family of homothetic Logarithmic Spirals.