Internal problem ID [10825]
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 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {x^{3} x^{\prime \prime }+1=0} \]
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 70
dsolve(x(t)^3*diff(x(t),t$2)+1=0,x(t), singsol=all)
\begin{align*} x \left (t \right ) = \frac {\sqrt {c_{1} \left (c_{1}^{2} c_{2}^{2}+2 c_{1}^{2} c_{2} t +c_{1}^{2} t^{2}-1\right )}}{c_{1}} \\ x \left (t \right ) = -\frac {\sqrt {c_{1} \left (c_{1}^{2} c_{2}^{2}+2 c_{1}^{2} c_{2} t +c_{1}^{2} t^{2}-1\right )}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.036 (sec). Leaf size: 58
DSolve[x[t]^3*x''[t]+1==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {\sqrt {-1+c_1{}^2 (t+c_2){}^2}}{\sqrt {c_1}} \\ x(t)\to \frac {\sqrt {-1+c_1{}^2 (t+c_2){}^2}}{\sqrt {c_1}} \\ \end{align*}