numerically solving nonlinear ODE which could be converted to abel equation of the second kind? I would like to solve this equation numerically, but none of the algorithms seems to fit this. I know that this equation could be transformed into abel equation of the second kind which has plenty ways of solving whether numerically or analytically, but I don't know how.
$ay''+by'+csin(y-nx)=0$
$y(0)=y'(0)=0$
 A: You can solve this using the forward-euler method (I think its called like this).
Basically you choose a step $\Delta x$. The smaller this value, the more accurate your result.
So for each step, you want to calculate $y'', y' and y$ and from that, you can go to the next iteration.
So basically you got the initial conditions for $y', y$ at $x=0$. If you plug this back into your differential equation, you can directly solve for y''(0).
What you do then is the following:
If you know $y''(x)$, you can approximate $y'(x+\Delta x)$ using the taylor-series like this:
$$y'(x+\Delta x) = y'(x) + y''(x)\cdot\Delta x$$
Doing the same for $y$:
$$y(x+\Delta x) = y(x) + y'(x)\cdot\Delta x$$
And you get $y''(x+\Delta x)$ by plugging $y'(x+\Delta x)$ and $y(x+\Delta x)$ back into your DE.
Note that the equation above are only an approximation. Thats why $\Delta x$ should be small.
A: Any ODE system with any degree of differentiation involved can be transformed to a first order system of a higher dimension. Just add components for all the derivatives involved, excepting the highest. Consequently, the standard numerical methods for solving ODE are constructed for first order systems.
Methods that start from a second order equation/system (Verlet, Beeman, Numerov) or from a partitioned system (symplectic integration)--which have a large overlap--are mostly for systems with structure, such as conservative mechanical systems where energy and other quantities are constants of motion. Then the methods can be designed to preserve that structure to a higher error order.
For the given equation, you can introduce an additional variable $v=y'$ to get the system
$$
\pmatrix{y'\\v'}
=
\pmatrix{v\\−(bv+c\sin(y−nx))/a}
$$
Alternatively, one could set $v=ay'+by$ so that $v'=-c\sin(y-nx)$ and together
$$
\pmatrix{y'\\v'}
=
\pmatrix{(v-by)/a\\−c\sin(y−nx)}
$$
There is no unique way to construct a first order system, it depends on the situation if there is a preferred first order formulation.
