23.4.292 problem 295

Internal problem ID [6594]
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 : 295
Date solved : Tuesday, September 30, 2025 at 03:26:44 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} a^{2} {y^{\prime \prime }}^{2}&=\left (1+{y^{\prime }}^{2}\right )^{3} \end{align*}

✓ Maple. Time used: 0.093 (sec). Leaf size: 93

ode:=a^2*diff(diff(y(x),x),x)^2 = (1+diff(y(x),x)^2)^3; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -i x +c_1 \\ y &= i x +c_1 \\ y &= \frac {\left (a +x +c_1 \right ) \left (-a +x +c_1 \right )}{\sqrt {-c_1^{2}-2 c_1 x +a^{2}-x^{2}}}+c_2 \\ y &= \frac {\left (a +x +c_1 \right ) \left (a -x -c_1 \right )}{\sqrt {-c_1^{2}-2 c_1 x +a^{2}-x^{2}}}+c_2 \\ \end{align*}

✓ Mathematica. Time used: 0.424 (sec). Leaf size: 141

ode=a^2*D[y[x],{x,2}]^2 == (1 + D[y[x],x]^2)^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_2-i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2}\\ y(x)&\to i \sqrt {a^2 \left (-1+c_1{}^2\right )-2 a c_1 x+x^2}+c_2\\ y(x)&\to c_2-i \sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}\\ y(x)&\to i \sqrt {a^2 \left (-1+c_1{}^2\right )+2 a c_1 x+x^2}+c_2 \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2*Derivative(y(x), (x, 2))**2 - (Derivative(y(x), x)**2 + 1)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-(a**2*Derivative(y(x), (x, 2))**2)**(1/3)/2 + sqrt(3)*I*(