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Numerical solution for a second-order differential equation
I am looking for the analytical solution and a numerical one of the following second-order differential equation
$$\frac{d^2y}{d^2x} + \frac{(y(x) – 0.5)d^2y}{d^2x} + Q(y(x)) = 0$$
where $Q(y)$ is an odd polynomial.
A:
If you use the ansatz $y(x)=\tan(\beta\,x)$ then $y$ is in fact a linear function of $x$. The differential equation can therefore be decoupled to two first-order equations and it is decidable how to integrate them.
If, however, you use the ansatz $y(x)=\cos(\alpha\,x)+\frac12\sin(\alpha\,x)$ then you have a separable equation. It is given by
$$\frac{d^2y}{d^2x}+\frac{\cot(\alpha\,x)d^2y}{d^2x}+\left(\frac{\sin(2\,\alpha\,x)}{\alpha\,\cos(\alpha\,x)}+\frac{\sin(2\,\alpha\,x)}{\alpha^2\,\cos(\alpha\,x)}+Q(y(x))\right)=0$$
The solution method consists in seeking for separable functions $y(x)=g(x)\,h(x)$. In this case this is easy, as you only have a linear problem for $h(x)$.
The main problem is that, as $\alpha$ is a free parameter, you have to guess the correct value of $\alpha$.
A:
You might be interested in What makes a good initial condition for an ODE?
As a specific example of a second-order differential
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