23.2.213 problem 219

Internal problem ID [5568]
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 : 219
Date solved : Tuesday, September 30, 2025 at 12:55:42 PM
CAS classification : [_rational]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.946 (sec). Leaf size: 131

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

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

✗ Sympy

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

Timed Out