14.5.8 problem Problem 8

Internal problem ID [3691]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 1, First-Order Differential Equations. Section 1.9, Exact Differential Equations. page 91
Problem number : Problem 8
Date solved : Tuesday, September 30, 2025 at 06:56:04 AM
CAS classification : [[_homogeneous, `class A`], _exact, _rational, _Riccati]

\begin{align*} \frac {1}{x}-\frac {y}{x^{2}+y^{2}}+\frac {x y^{\prime }}{x^{2}+y^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 12

ode:=1/x-y(x)/(x^2+y(x)^2)+x/(x^2+y(x)^2)*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\tan \left (\ln \left (x \right )+c_1 \right ) x \]

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 15

ode=(1/x-y[x]/(x^2+y[x]^2))+x/(x^2+y[x]^2)*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x \tan (-\log (x)+c_1) \end{align*}

✓ Sympy. Time used: 0.187 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = \frac {x \left (- i C_{1} - i e^{2 i \log {\left (x \right )}}\right )}{C_{1} - e^{2 i \log {\left (x \right )}}} \]