83.23.3 problem 3

Internal problem ID [19213]
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 : 3
Date solved : Monday, March 31, 2025 at 06:57:30 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Maple. Time used: 0.097 (sec). Leaf size: 60

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

\begin{align*} y &= \sqrt {-m \left (x^{2}-1\right )} \\ y &= -\sqrt {\left (-x^{2}+1\right ) m} \\ y &= c_1 x -\sqrt {c_1^{2}+m} \\ y &= c_1 x +\sqrt {c_1^{2}+m} \\ \end{align*}

✓ Mathematica. Time used: 13.914 (sec). Leaf size: 101

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

\begin{align*} y(x)\to -\sqrt {m} \sec \left (c_1 \sqrt {m}\right ) \left (-1+x \sin \left (c_1 \sqrt {m}\right )\right ) \\ y(x)\to -\sqrt {m} \sec \left (c_1 \sqrt {m}\right ) \left (1+x \sin \left (c_1 \sqrt {m}\right )\right ) \\ y(x)\to -\sqrt {m-m x^2} \\ y(x)\to \sqrt {m-m x^2} \\ \end{align*}

✗ Sympy

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

Timed Out