66.2.10 problem Problem 10

Internal problem ID [13909]
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
Date solved : Tuesday, January 28, 2025 at 06:08:15 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{3} x^{\prime \prime }+1&=0 \end{align*}

✓ Solution by Maple

Time used: 0.119 (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: 2.844 (sec). Leaf size: 93

DSolve[x[t]^3*D[x[t],{t,2}]+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*}