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?
A question about homogeneous ODE's and the usefulness of certain solutions
-
0I know this may involve effort, but can you give an example of this happening? It will reduce the vagueness of the question, and help others answer it better. – 2017-01-12
-
1Certainly. Consider the equation $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x+y}{x-y}$, with this one can arrive to the point where $\arctan\left(\frac{y}{x}\right) = \ln\left(\sqrt{x^2+y^2}\right) + C$. I doubt one can solve for $y$ here. Is there any context in which one is able to acquire useful information from the second equation? Just some exampel based on theoretical circumstances would do. – 2017-01-12
-
0Thank you for the example, this function is moderately difficult to even sketch at a glance. – 2017-01-12
-
1In many real world applications, we don't always want a solution to an equation, but we would like to know something about it (what happens as it tends to infinity, the flux of the solution through a surface, what happens when we move our solution, ect) and knowing as much as we can about the DE is generally helpful. However, if your only goal is to graph it, it is usually easier to use Euler's method on a computer than to solve. – 2017-01-12
-
1This is still gives useful information about the relationship between $x$ and $y$, and perhaps in a form that is easier to understand than the original ODE. It is usually hard, for example, to find discontinuities based on just the ODE, but an implicit definition makes that problem easier. Consider the implicit plot shown here: https://www.wolframalpha.com/input/?i=ImplicitPlot%5Barctan(y%2Fx)%3D%3Dln(sqrt(x%5E2%2By%5E2)),%7Bx,-1,1%7D,%7By,-1,1%7D%5D – 2017-01-12
-
0I was primarily looking for someone to tell me that one is still theoretically able to obtain useful information from this so I suppose these comments will suffice. Thank you. – 2017-01-12
-
0@David I believe you should add your example to your question, it was clear to me what you were asking, but it may raise some doubts on others. – 2017-01-12
-
2@David Sure it is possible. Sometimes it is even possible to get more interesting insights without finding solutions at all. That's what qualitative theory of ODEs and dynamical systems is all about :) – 2017-01-12
1 Answers
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.