83.23.23 problem 23

Internal problem ID [19233]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 23
Date solved : Monday, March 31, 2025 at 07:00:04 PM
CAS classification : [_rational]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 1.963 (sec). Leaf size: 123

ode=a*x*y[x]*D[y[x],x]^2+(x^2-a*y[x]^2-b)*D[y[x],x]-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 {b}{1+a c_1}\right )} \\ y(x)\to -\sqrt {-\frac {\left (\sqrt {b}+x\right )^2}{a}} \\ y(x)\to \sqrt {-\frac {\left (\sqrt {b}+x\right )^2}{a}} \\ y(x)\to -\sqrt {-\frac {\left (\sqrt {b}-x\right )^2}{a}} \\ y(x)\to \sqrt {-\frac {\left (\sqrt {b}-x\right )^2}{a}} \\ \end{align*}

✗ Sympy

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

Timed Out