Second-order ODE involving two functionsI am wondering how to find a general analytical solution to the following ODE:dydtd2xdt2=dxdtd2ydt2The solution method might be relatively simple; but right now I don't know how to approach this problem.

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Second-order ODE involving two functionsI am wondering how to find a general analytical solution to the following ODE:dydtd2xdt2=dxdtd2ydt2The solution method might be relatively simple; but right now I don&#039;t know how to approach this problem.

Charlie Haley · Accepted Answer

Step 1I find it more convenient to rewrite the equation using Newton&#039;s notation. Instead of writing dxdt, it is more helpful to write x&#039;. Thus, the equation isx′y″=x″y′.Now, suppose x′=0. Then x0 is trivial, so every differentiable function y∈RR satisfies the equation. This holds analogously if y′=0. Otherwise, we can divide by x&#039;y&#039;, thusy″y′=x″x′.Step 2There are four cases to consider from here: x′&amp;lt;0 and y′&amp;lt;0;x′&amp;lt;0 and y′&amp;gt;0;x′&amp;gt;0 and y′&amp;lt;0; and x′&amp;gt;0 and y′&amp;gt;0. These cases simplify the equation respectivelyln⁡(−x′)+C=ln⁡(−y′)ln⁡(−x′)+C=ln⁡(y′)ln⁡(x′)+C=ln⁡(−y′)ln⁡(x′)+C=ln⁡(y′)which are equivalent toeCx′=y′−eCx′=y′−eCx′=y′eCx′=y′.These cases simply reduce to y′=Ax′where A≠0. Therefore, we have that y(t)=Ax(t)+B, where A≠0. Remember that this is in the case when neither x nor y is a constant function.

Second-order ODE involving two functions I am wondering how

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