23.2.170 problem 173

Internal problem ID [5525]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 173
Date solved : Tuesday, September 30, 2025 at 12:50:28 PM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(y)]`]]

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

✓ Maple. Time used: 0.655 (sec). Leaf size: 51

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

\begin{align*} y &= \sqrt {a^{2}-x^{2}} \\ y &= -\sqrt {a^{2}-x^{2}} \\ y &= c_{1} x^{2}-c_{1} a^{2}-\frac {1}{4 c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 0.263 (sec). Leaf size: 67

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

\begin{align*} y(x)&\to \frac {a^2-x^2+c_1{}^2}{2 c_1}\\ y(x)&\to \text {Indeterminate}\\ y(x)&\to -\sqrt {a^2-x^2}\\ y(x)&\to \sqrt {a^2-x^2} \end{align*}

✗ Sympy

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

Timed Out