I want to solve (in parametric fashion and NOT numerically) the following nonlinear second-order ODE.
$\frac{\displaystyle 1}{\displaystyle m}w'' = A + B - C^2 \frac{\displaystyle (w'-n)}{\displaystyle(C-G)(w'-n)+mw} + \frac{\displaystyle2C(w'-n)}{\displaystyle m}$
where $A,B,C,n,m,G$ are constants.
I did try to parse the expression for the second-order ODE in Matlab LiveScript utilizing MUPAD in the background. Is it possible to provide a format that can be parsed efficiently in MUPAD? I am new to solving ODEs with software and I do not understand what the problem might be. I give the outline of the code:
syms A B C G m n
syms w(t)
diff_equation = 1/m * diff(w(t), t, 2) - A - B + (C^2 * (diff(w(t), t) - n))/((C - G) * (diff(w(t), t) - n) - mw) - (2 * C * (diff(w(t), t) - n))/m
syms v0 w0
init_condition = {w(sym(0)) == w0, subs(diff(w(t), t), t, sym(0)) == v0}
aux1 = unique([diff_equation, init_condition{:}]);
dsolve(symfun(aux1, symvar(w(t))))
If I remove the term $mw$ from the denominator of the first fraction that appears, then I can get a solution, otherwise, MUPAD cannot provide a parametric expression of the solution.
Thank you.
Probably this system will have closed form solutions only for very specific values of the constants. You can consider the original system \begin{align} \displaystyle \frac{1}{m}\frac{dz}{dt} &= A + B - C^2\frac{(z-n)}{(C-G)(z-n) + mw} + 2C(z-n)/m\\ \displaystyle \frac{dw}{dt} &= z \end{align} and try to find first integrals for it. Dividing the second equation by the first you will get, after some simplifications: \begin{align} \displaystyle \frac{dw}{dz} &= \frac{mzw-P_1(z)}{P_2(z)w+P_3(z)},\\ \end{align} where $P_i$ are polynomials with $deg(P_1)=2$, $deg(P_2)=1$ and $deg(P_3)=2$. By making the substitution $$w=\frac{1}{f(z)}-\frac{P_3(z)}{P_2(z)}$$ you find \begin{align} \displaystyle \frac{df}{dz} &= R_1(z)f^3+R_2(z)f^2,\\ \end{align} where $R_1(z)$ and $R_2(z)$ are rational functions. This is a Abel equation of the first kind. For some values of parameters this equation can be made separable, Bernoulli, etc.