Is it true that the number of arbitrary constants in the solution (if solutions exist) always equal to the order of an ordinary differential equation? If yes, how to "prove" such a statement, if it can be proved at all? If no, what are typical counterexamples?
[Math] Is it true that the number of arbitrary constants in the solution always equal to order of the ordinary differential equation
ordinary differential equations
Best Answer
The statement is indeed true. I don't know how one can prove it rigorously. However, here is some further insight:
If you have with you a general equation in x, y, or more variables with some constants a, b, ... Suppose there are n constants in the equation.
What would you do if you wanted to find the differential equation that this curve satisfies? You would differentiate it w.r.t. x some number of times. If you differentiate it once, you get a new equation (the equation of the curve is also an equation). If you differentiate it twice, you get another. In general, if you differentiate k times, you will have k+1 equations. Our knowledge of linear algebra tells us that generally, one needs n+1 equations to eliminate n variables. How can we get n+1 equations? Differentiate it once, twice... n times. If you try to eliminate all the constants, you will be left with a final differential equation of order n.
Hope this helps!