Internal problem ID [10149]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 6, non-linear second order
Problem number: 1827 (book 6.236).
ODE order: 2.
ODE degree: 2.
CAS Maple gives this as type [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]
\[ \boxed {{y^{\prime \prime }}^{2}-a y=b} \]
✓ Solution by Maple
Time used: 0.046 (sec). Leaf size: 206
dsolve(diff(diff(y(x),x),x)^2-a*y(x)-b=0,y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {b}{a} \\ \sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ -\sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \sqrt {3}\, a \left (\int _{}^{y \left (x \right )}\frac {1}{\sqrt {-a \left (4 \textit {\_a} \sqrt {a \textit {\_a} +b}\, a +4 \sqrt {a \textit {\_a} +b}\, b -c_{1} \right )}}d \textit {\_a} \right )-x -c_{2} &= 0 \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.116 (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 ) \operatorname {Hypergeometric2F1}\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 ) \operatorname {Hypergeometric2F1}\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*}