83.22.15 problem 15

Internal problem ID [19198]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 15
Date solved : Monday, March 31, 2025 at 06:53:46 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} {y^{\prime }}^{2} \left (-a^{2}+x^{2}\right )-2 y^{\prime } x y+y^{2}+a^{4}&=0 \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 80

ode:=diff(y(x),x)^2*(-a^2+x^2)-2*x*y(x)*diff(y(x),x)+y(x)^2+a^4 = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {-a^{2}+x^{2}}\, a \\ y &= -\sqrt {-a^{2}+x^{2}}\, a \\ y &= c_1 x -\sqrt {c_1^{2} a^{2}-a^{4}} \\ y &= c_1 x +\sqrt {c_1^{2} a^{2}-a^{4}} \\ \end{align*}

✓ Mathematica. Time used: 0.479 (sec). Leaf size: 101

ode=D[y[x],x]^2*(x^2-a^2)-2*D[y[x],x]*x*y[x]+y[x]^2+a^4==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 x-\sqrt {-a^4+a^2 c_1{}^2} \\ y(x)\to \sqrt {-a^4+a^2 c_1{}^2}+c_1 x \\ y(x)\to -\sqrt {a^2 x^2-a^4} \\ y(x)\to \sqrt {a^2 x^2-a^4} \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**4 - 2*x*y(x)*Derivative(y(x), x) + (-a**2 + x**2)*Derivative(y(x), x)**2 + y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out