Second Order ODE with Power

Hello,

I have this form of equation

x'' = A/x^2 *(B+C*(x')^2+C*(x')^4)

I wrote this script

syms x(t) A B C D vb b

v=diff(x,t,2)==(A/x)*(B+C*(diff(x,t))^2+(C*(diff(x,t))^4);

Dx=diff(x,t);

initial = [x(0)==b, Dx(0)==vb];

xSol(t) = dsolve(v,initial)

But I had this output

Warning: Unable to find explicit solution.

xSol(t) =
 
[ empty sym ]

I thought of solving it to some extent and apply numerical methods. I later came up with an equation of the form

integral ((A+B*X^a)/(C+D*X^a))dx, please note that constants A, B, C, and D here are different from the ones above.

This, I believe is a form of hypergeometric expression. I don't know how to move further from here.

Thank you.

Answer 1

[Star Strider]

The best way to integrate it numerically is something like this:

syms x(t) A B C D vb b Y t 
v=diff(x,t,2)==(A/x)*(B+C*(diff(x,t))^2+(C*(diff(x,t))^4));
[VF,Subs] = odeToVectorField(v);
odefcn = matlabFunction(VF, 'Vars',{t,Y,A,B,C,D})

A = ...;
B = ...;
C = ...;
D = ...;

tspan = ...;
b = ...;
vb = ...;
ic = [b; vb];

[T,X] = ode45(@(t,Y)odefcn(t,Y,A,B,C,D), tspan, ic);

figure
plot(T,X)
legend(string(Subs))

I chose ode45 here because it usually works. If the constants vary by several orders-of-magnitude, the equation would likely be ‘stiff’ and a different solver, such as ode15s would be necessary.

[Shozeal]

Thank you, Strider.

This line here [VF,Subs] = odeToVectorField(v); is what I've been looking for.

[Star Strider]

As always, my pleasure!

I am happy to have helped you find it! It and matlabFucntion will allow you to create the function the numeric differential funciton integrators need.