Internal problem ID [9405]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1826.
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
Solve \begin {gather*} \boxed {\left (y^{\prime \prime }\right )^{2}-a y-b=0} \end {gather*}
✓ Solution by Maple
Time used: 0.043 (sec). Leaf size: 201
dsolve(diff(diff(y(x),x),x)^2-a*y(x)-b=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {b}{a} \\ \int _{}^{y \relax (x )}\frac {a \sqrt {3}}{\sqrt {a \left (4 \textit {\_a} \sqrt {\textit {\_a} a +b}\, a +4 \sqrt {\textit {\_a} a +b}\, b -c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {a \sqrt {3}}{\sqrt {a \left (4 \textit {\_a} \sqrt {\textit {\_a} a +b}\, a +4 \sqrt {\textit {\_a} a +b}\, b -c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}-\frac {3 a}{\sqrt {-3 a \left (4 \textit {\_a} \sqrt {\textit {\_a} a +b}\, a +4 \sqrt {\textit {\_a} a +b}\, b -c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \int _{}^{y \relax (x )}\frac {3 a}{\sqrt {-3 a \left (4 \textit {\_a} \sqrt {\textit {\_a} a +b}\, a +4 \sqrt {\textit {\_a} a +b}\, b -c_{1}\right )}}d \textit {\_a} -x -c_{2} = 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.626 (sec). Leaf size: 201
DSolve[-b - a*y[x] + y''[x]^2 == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} \text {Solve}\left [\frac {(a y(x)+b)^2 \left (1-\frac {4 (a y(x)+b)^{3/2}}{3 a c_1}\right ) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};\frac {4 (b+a y(x))^{3/2}}{3 a c_1}\right ){}^2}{a^2 \left (-\frac {4 (a y(x)+b)^{3/2}}{3 a}+c_1\right )}=(x+c_2){}^2,y(x)\right ] \\ \text {Solve}\left [\frac {(a y(x)+b)^2 \left (1+\frac {4 (a y(x)+b)^{3/2}}{3 a c_1}\right ) \, _2F_1\left (\frac {1}{2},\frac {2}{3};\frac {5}{3};-\frac {4 (b+a y(x))^{3/2}}{3 a c_1}\right ){}^2}{a^2 \left (\frac {4 (a y(x)+b)^{3/2}}{3 a}+c_1\right )}=(x+c_2){}^2,y(x)\right ] \\ \end{align*}