44.8.13 problem 2(e)

Internal problem ID [9215]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 1. What is a differential equation. Problems for Review and Discovery. Page 53
Problem number : 2(e)
Date solved : Tuesday, September 30, 2025 at 06:14:54 PM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 30

ode:=diff(y(x),x) = (x+y(x))/(x-y(x)); 
ic:=[y(1) = 1]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \tan \left (\operatorname {RootOf}\left (4 \textit {\_Z} -2 \ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )-4 \ln \left (x \right )+2 \ln \left (2\right )-\pi \right )\right ) x \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 46

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

\[ \text {Solve}\left [\frac {1}{2} \log \left (\frac {y(x)^2}{x^2}+1\right )-\arctan \left (\frac {y(x)}{x}\right )=\frac {1}{4} (2 \log (2)-\pi )-\log (x),y(x)\right ] \]

✓ Sympy. Time used: 0.639 (sec). Leaf size: 32

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

\[ \log {\left (x \right )} = - \log {\left (\sqrt {1 + \frac {y^{2}{\left (x \right )}}{x^{2}}} \right )} + \operatorname {atan}{\left (\frac {y{\left (x \right )}}{x} \right )} - \frac {\pi }{4} + \frac {\log {\left (2 \right )}}{2} \]