23.4.57 problem 57

Internal problem ID [6359]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 57
Date solved : Tuesday, September 30, 2025 at 02:52:29 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 96

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

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

✓ Mathematica. Time used: 8.332 (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))) + De