23.4.233 problem 233

Internal problem ID [6535]
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 : 233
Date solved : Friday, October 03, 2025 at 02:09:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.145 (sec). Leaf size: 55

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

\begin{align*} y &= 0 \\ y &= c_2 \,{\mathrm e}^{-\int \frac {x \sqrt {a^{2}-x^{2}}}{-a^{2} b +b \,x^{2}+c_1 \sqrt {a^{2}-x^{2}}}d x} \\ \end{align*}

✓ Mathematica. Time used: 10.446 (sec). Leaf size: 53

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

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

✗ Sympy

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

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