60.7.87 problem 1698 (book 6.107)

Internal problem ID [11637]
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 : Sunday, March 30, 2025 at 08:32:16 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*}

✓ Maple. Time used: 0.021 (sec). Leaf size: 39

ode:=diff(diff(y(x),x),x)*y(x)+diff(y(x),x)^2-a = 0; 
dsolve(ode,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*}

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

ode=-a + D[y[x],x]^2 + y[x]*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a + y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(a - y(x)*Derivative(y(x), (x, 2))) + Derivative(y(x), x) cannot be solved by the factorable group method