60.7.209 problem 1800 (book 6.209)

Internal problem ID [11798]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1800 (book 6.209)
Date solved : Monday, January 27, 2025 at 11:36:00 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{3} y^{\prime \prime }-a&=0 \end{align*}

✓ Solution by Maple

Time used: 0.085 (sec). Leaf size: 46

dsolve(y(x)^3*diff(diff(y(x),x),x)-a=0,y(x), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {c_{1} \left (\left (x +c_{2} \right )^{2} c_{1}^{2}+a \right )}}{c_{1}} \\ y &= -\frac {\sqrt {c_{1} \left (\left (x +c_{2} \right )^{2} c_{1}^{2}+a \right )}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.776 (sec). Leaf size: 63

DSolve[-a + y[x]^3*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\sqrt {a+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}} \\ y(x)\to \frac {\sqrt {a+c_1{}^2 (x+c_2){}^2}}{\sqrt {c_1}} \\ y(x)\to \text {Indeterminate} \\ \end{align*}