69.1.61 problem 80

Internal problem ID [14143]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 80
Date solved : Monday, March 31, 2025 at 12:10:37 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _exact, _rational]

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.123 (sec). Leaf size: 95

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {x}{x^2+K[2]^2}+K[2]-\int _1^x\left (\frac {2 K[2]^2}{\left (K[1]^2+K[2]^2\right )^2}-\frac {1}{K[1]^2+K[2]^2}\right )dK[1]\right )dK[2]+\int _1^x\left (K[1]-\frac {y(x)}{K[1]^2+y(x)^2}\right )dK[1]=c_1,y(x)\right ] \]

✗ Sympy

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

Timed Out