7.3.16 problem 18

Internal problem ID [2333]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.4. Page 24
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 05:28:57 AM
CAS classification : [[_homogeneous, `class A`], _rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 24

ode:=diff(y(t),t) = (t+y(t))/(t-y(t)); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \tan \left (\operatorname {RootOf}\left (-2 \textit {\_Z} +\ln \left (\sec \left (\textit {\_Z} \right )^{2}\right )+2 \ln \left (t \right )+2 c_1 \right )\right ) t \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 36

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

\[ \text {Solve}\left [\frac {1}{2} \log \left (\frac {y(t)^2}{t^2}+1\right )-\arctan \left (\frac {y(t)}{t}\right )=-\log (t)+c_1,y(t)\right ] \]

✓ Sympy. Time used: 0.626 (sec). Leaf size: 26

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

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