Internal problem ID [12173]
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]]
\[ \boxed {x^{3} x^{\prime \prime }=-1} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 52
dsolve(x(t)^3*diff(x(t),t$2)+1=0,x(t), singsol=all)
\begin{align*} x \left (t \right ) &= \frac {\sqrt {\left (1+c_{1} \left (c_{2} +t \right )\right ) \left (-1+c_{1} \left (c_{2} +t \right )\right ) c_{1}}}{c_{1}} \\ x \left (t \right ) &= -\frac {\sqrt {\left (1+c_{1} \left (c_{2} +t \right )\right ) \left (-1+c_{1} \left (c_{2} +t \right )\right ) c_{1}}}{c_{1}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 4.287 (sec). Leaf size: 93
DSolve[x[t]^3*x''[t]+1==0,x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\frac {\sqrt {c_1{}^2 t^2+2 c_2 c_1{}^2 t-1+c_2{}^2 c_1{}^2}}{\sqrt {c_1}} \\ x(t)\to \frac {\sqrt {c_1{}^2 t^2+2 c_2 c_1{}^2 t-1+c_2{}^2 c_1{}^2}}{\sqrt {c_1}} \\ x(t)\to \text {Indeterminate} \\ \end{align*}