Internal problem ID [10843]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 40(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {m x^{\prime \prime }-f \left (x\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 62
dsolve(m*diff(x(t),t$2)=f(x(t)),x(t), singsol=all)
\begin{align*} \int _{}^{x \left (t \right )}\frac {m}{\sqrt {m \left (c_{1} m +2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )\right )}}d \textit {\_b} -t -c_{2} = 0 \\ \int _{}^{x \left (t \right )}-\frac {m}{\sqrt {m \left (c_{1} m +2 \left (\int f \left (\textit {\_b} \right )d \textit {\_b} \right )\right )}}d \textit {\_b} -t -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 44
DSolve[m*x''[t]==f[x[t]],x[t],t,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{x(t)}\frac {1}{\sqrt {c_1+2 \int _1^{K[2]}\frac {f(K[1])}{m}dK[1]}}dK[2]{}^2=(t+c_2){}^2,x(t)\right ] \]