83.6.1 problem 1

Internal problem ID [19046]
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. Exercise II (E) at page 19
Problem number : 1
Date solved : Monday, March 31, 2025 at 06:36:42 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _exact, _rational]

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

✓ Maple. Time used: 0.058 (sec). Leaf size: 34

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

\[ y = x \cot \left (\operatorname {RootOf}\left (2 c_1 \,a^{2} \sin \left (\textit {\_Z} \right )^{2}-2 \textit {\_Z} \,a^{2} \sin \left (\textit {\_Z} \right )^{2}-x^{2}\right )\right ) \]

✓ Mathematica. Time used: 0.218 (sec). Leaf size: 33

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

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

✗ Sympy

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

Timed Out