54.7.211 problem 1837 (book 6.246)

Internal problem ID [13060]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1837 (book 6.246)
Date solved : Wednesday, October 01, 2025 at 03:31:50 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \sqrt {a {y^{\prime \prime }}^{2}+b {y^{\prime }}^{2}}+c y y^{\prime \prime }+d {y^{\prime }}^{2}&=0 \end{align*}

✗ Maple

ode:=(a*diff(diff(y(x),x),x)^2+b*diff(y(x),x)^2)^(1/2)+c*y(x)*diff(diff(y(x),x),x)+d*diff(y(x),x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=d*D[y[x],x]^2 + c*y[x]*D[y[x],{x,2}] + Sqrt[b*D[y[x],x]^2 + a*D[y[x],{x,2}]^2] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(2)*sqrt(b/d**2 - 2*c*y(x)*Derivative(y(x), (x, 2))/d + sqrt(4*a*d**2*Derivative(y(x), (x, 2))**2 + b**2 - 4*b*c*d*y(x)*Derivative(y(x), (x, 2)))/d**2)/2 + Derivative(y(x), x) cannot be solved by the factorable group method