Numerical method to solve high order ODEs.

[meridian]

here is a simplified version of my working equtions
<br /> y&#039;&#039;&#039; = \frac{(y&#039;&#039; y+y&#039; y) y + y&#039;y&#039;&#039;}{y&#039; + y&#039;&#039;}<br />
and 3 related boundary conditions, is there some hints to solve such equation numerically?

ThX

 

[Defennder]

This might sound crazy, but note that everything here is a function of y. You may not have to resort to numerical solutions. But then again I haven't tried it out yet. Looks a little intimidating.

 

[meridian]

in the equation,
y = y(x)
The original equation are much more complex, it is not possible to get a exact analytical solution for that. What I want to learn is the general numerical method to solve such equation.

 

[HallsofIvy]

Let u= y'(x), v= y"(x). Then your equation is becomes
y&#039;= \frac{(v y+u y) y + uv}{y + v}
That together with y'= u and u'= v gives you three interconnected first order equations. Do, say, a 4th order Runge-Kutta, advancing the step in all three equations at the same time.

 

[Topher925]

Do, say, a 4th order Runge-Kutta, advancing the step in all three equations at the same time.

Thats would I would suggest. Although the Adams-Moulton-Bashforth method would work as well and probably give you a little bit more accuracy. But you will need the RK4 for the first couple steps anyway.