60.7.107 problem 1698 (book 6.107)

Internal problem ID [11696]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1698 (book 6.107)
Date solved : Monday, January 27, 2025 at 11:30:11 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime } y+{y^{\prime }}^{2}-a&=0 \end{align*}

✓ Solution by Maple

Time used: 0.148 (sec). Leaf size: 39

dsolve(diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2-a=0,y(x), singsol=all)
 

\begin{align*} y &= \sqrt {a \,x^{2}-2 c_{1} x +2 c_{2}} \\ y &= -\sqrt {a \,x^{2}-2 c_{1} x +2 c_{2}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 14.033 (sec). Leaf size: 117

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

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