54.7.49 problem 1653 (book 6.62)

Internal problem ID [12898]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1653 (book 6.62)
Date solved : Friday, October 03, 2025 at 03:50:39 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }&=a \sqrt {{y^{\prime }}^{2}+b y^{2}} \end{align*}

✓ Maple. Time used: 0.141 (sec). Leaf size: 41

ode:=diff(diff(y(x),x),x) = a*(diff(y(x),x)^2+b*y(x)^2)^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\int \operatorname {RootOf}\left (x -\int _{}^{\textit {\_Z}}-\frac {1}{-a \sqrt {\textit {\_f}^{2}+b}+\textit {\_f}^{2}}d \textit {\_f} +c_1 \right )d x +c_2} \\ \end{align*}

✓ Mathematica. Time used: 0.179 (sec). Leaf size: 76

ode=D[y[x],{x,2}]==a*Sqrt[D[y[x],x]^2+b*y[x]^2]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int \frac {\text {$\#$1}}{K[1] \left (\frac {\text {$\#$1}^2}{K[1]^2}-a \sqrt {\frac {\text {$\#$1}^2}{K[1]^2}+b}\right )}d\frac {\text {$\#$1}}{K[1]}\&\right ][c_1-\log (K[1])]}dK[1]=x-c_2,y(x)\right ] \]

✗ Sympy

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

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