54.7.44 problem 1646 (6.56)

Internal problem ID [12893]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1646 (6.56)
Date solved : Wednesday, October 01, 2025 at 02:45:02 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} y^{\prime \prime }+a y \left ({y^{\prime }}^{2}+1\right )^{2}&=0 \end{align*}

✓ Maple. Time used: 0.022 (sec). Leaf size: 96

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

\begin{align*} a \int _{}^{y}\frac {\textit {\_a}^{2}+2 c_1}{\sqrt {-\left (-1+a \left (\textit {\_a}^{2}+2 c_1 \right )\right ) a \left (\textit {\_a}^{2}+2 c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\ -a \int _{}^{y}\frac {\textit {\_a}^{2}+2 c_1}{\sqrt {-\left (-1+a \left (\textit {\_a}^{2}+2 c_1 \right )\right ) a \left (\textit {\_a}^{2}+2 c_1 \right )}}d \textit {\_a} -x -c_2 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 8.41 (sec). Leaf size: 816

ode=a*y[x]*(1 + D[y[x],x]^2)^2 + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*(Derivative(y(x), x)**2 + 1)**2*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-1 + sqrt(-a*y(x)*Derivative(y(x), (x, 2)))/(a*y(x))) + Derivative(y(x), x) cannot be solved by the factorable group method