23.4.130 problem 130

Internal problem ID [6432]
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 : 130
Date solved : Tuesday, September 30, 2025 at 02:56:44 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 62

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

\begin{align*} y &= 0 \\ y &= \frac {\sqrt {-a \left (-c_2 \cos \left (2 a x \right )+c_1 \sin \left (2 a x \right )\right )}}{a} \\ y &= -\frac {\sqrt {-a \left (-c_2 \cos \left (2 a x \right )+c_1 \sin \left (2 a x \right )\right )}}{a} \\ \end{align*}

✓ Mathematica. Time used: 0.844 (sec). Leaf size: 23

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

\begin{align*} y(x)&\to c_2 \sqrt {\cos (a (2 x-c_1))} \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(-(2*a**2*y(x) + Derivative(y(x), (x, 2)))*y(x)) + Derivati