77.8.1 problem 1

Internal problem ID [20412]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Misc examples on chapter II at page 25
Problem number : 1
Date solved : Sunday, October 12, 2025 at 05:37:45 AM
CAS classification : [[_1st_order, _with_linear_symmetries]]

\begin{align*} \frac {x +y y^{\prime }}{x y^{\prime }-y}&=\sqrt {\frac {a^{2}-x^{2}-y^{2}}{x^{2}+y^{2}}} \end{align*}

✓ Maple. Time used: 0.064 (sec). Leaf size: 152

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

\[ -\frac {\arctan \left (\frac {a^{2}-2 x^{2}-2 y^{2}}{2 \sqrt {\left (a^{2}-x^{2}-y^{2}\right ) \left (x^{2}+y^{2}\right )}}\right ) \left (x^{2}+y^{2}\right ) \sqrt {\frac {a^{2}-x^{2}-y^{2}}{x^{2}+y^{2}}}+2 \sqrt {\left (a^{2}-x^{2}-y^{2}\right ) \left (x^{2}+y^{2}\right )}\, \left (c_1 -\arctan \left (\frac {x}{y}\right )\right )}{2 \sqrt {\left (a^{2}-x^{2}-y^{2}\right ) \left (x^{2}+y^{2}\right )}} = 0 \]

✓ Mathematica. Time used: 10.619 (sec). Leaf size: 116

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

\[ \text {Solve}\left [-\frac {a \sqrt {-\frac {-a^2+x^2+y(x)^2}{a^2}} \arcsin \left (\frac {\sqrt {x^2+y(x)^2}}{a}\right )+\sqrt {x^2+y(x)^2} \sqrt {-\frac {-a^2+x^2+y(x)^2}{x^2+y(x)^2}} \arctan \left (\frac {x}{y(x)}\right )}{\sqrt {-a^2+x^2+y(x)^2}}=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out