60.7.104 problem 1695 (book 6.104)

Internal problem ID [11693]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1695 (book 6.104)
Date solved : Monday, January 27, 2025 at 11:30:09 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Solution by Maple

Time used: 0.172 (sec). Leaf size: 53

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

\begin{align*} \int _{}^{y}\frac {1}{\sqrt {2 a \ln \left (\textit {\_a} \right )-c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ -\int _{}^{y}\frac {1}{\sqrt {2 a \ln \left (\textit {\_a} \right )-c_{1}}}d \textit {\_a} -x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.181 (sec). Leaf size: 111

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

\begin{align*} y(x)\to \exp \left (-\frac {2 a \text {erf}^{-1}\left (-i \sqrt {\frac {2}{\pi }} \sqrt {a e^{\frac {c_1}{a}} (x+c_2){}^2}\right ){}^2+c_1}{2 a}\right ) \\ y(x)\to \exp \left (-\frac {2 a \text {erf}^{-1}\left (i \sqrt {\frac {2}{\pi }} \sqrt {a e^{\frac {c_1}{a}} (x+c_2){}^2}\right ){}^2+c_1}{2 a}\right ) \\ \end{align*}