74.7.31 problem 31

Internal problem ID [16029]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 31
Date solved : Thursday, March 13, 2025 at 07:20:01 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} y^{\prime }&=\frac {1}{\frac {2 y \,{\mathrm e}^{-\frac {t}{y}}}{t}+\frac {t}{y}} \end{align*}

✓ Maple. Time used: 0.741 (sec). Leaf size: 31

ode:=diff(y(t),t) = 1/(2*y(t)*exp(-t/y(t))/t+t/y(t)); 
dsolve(ode,y(t), singsol=all);
 

\[ y = t \,{\mathrm e}^{-\operatorname {RootOf}\left (-2 \textit {\_Z} -{\mathrm e}^{{\mathrm e}^{\textit {\_Z}}+\textit {\_Z}}+{\mathrm e}^{{\mathrm e}^{\textit {\_Z}}}+2 \ln \left (t \right )+2 c_{1} \right )} \]

✓ Mathematica. Time used: 0.262 (sec). Leaf size: 43

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

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

✓ Sympy. Time used: 1.175 (sec). Leaf size: 20

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

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