23.4.301 problem 304

Internal problem ID [6603]
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 : 304
Date solved : Tuesday, September 30, 2025 at 03:27:25 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 1.728 (sec). Leaf size: 124

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

\begin{align*} y &= \frac {\operatorname {csgn}\left (b \right ) a \,\operatorname {csgn}\left (\sec \left (\frac {c_{1} -x}{a}\right )\right ) \sin \left (\frac {c_{1} -x}{a}\right )}{b} \\ y &= -\frac {\operatorname {csgn}\left (b \right ) a \,\operatorname {csgn}\left (\sec \left (\frac {c_{1} -x}{a}\right )\right ) \sin \left (\frac {c_{1} -x}{a}\right )}{b} \\ y &= \frac {a}{b} \\ y &= -\frac {a}{b} \\ y &= c_{1} \\ y &= \frac {a \left ({\mathrm e}^{\frac {\sqrt {b^{2} c_{1}^{2}-1}\, \left (c_2 +x \right )}{a}}-c_{1} \right )}{\sqrt {b^{2} c_{1}^{2}-1}} \\ \end{align*}

✗ Mathematica

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

{}

✗ Sympy

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

NotImplementedError : The given ODE -sqrt(2)*sqrt(2*y(x)*Derivative(y(x), (x, 2)) + sqrt(4*a**2*b**2