85.33.53 problem 53

Internal problem ID [22676]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 53
Date solved : Thursday, October 02, 2025 at 09:06:13 PM
CAS classification : [_separable]

\begin{align*} r^{\prime }&=\frac {r \left (1+\ln \left (t \right )\right )}{t \left (1+\ln \left (r\right )\right )} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 45

ode:=diff(r(t),t) = r(t)*(1+ln(t))/t/(1+ln(r(t))); 
dsolve(ode,r(t), singsol=all);
 

\begin{align*} r &= {\mathrm e}^{-1-\sqrt {1+\ln \left (t \right )^{2}+2 \ln \left (t \right )+2 c_1}} \\ r &= {\mathrm e}^{-1+\sqrt {1+\ln \left (t \right )^{2}+2 \ln \left (t \right )+2 c_1}} \\ \end{align*}

✓ Mathematica. Time used: 0.27 (sec). Leaf size: 38

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

\begin{align*} r(t)&\to \text {InverseFunction}\left [\frac {\log ^2(\text {$\#$1})}{2}+\log (\text {$\#$1})\&\right ]\left [\frac {\log ^2(t)}{2}+\log (t)+c_1\right ]\\ r(t)&\to 0 \end{align*}

✓ Sympy. Time used: 1.029 (sec). Leaf size: 42

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

\[ \left [ r{\left (t \right )} = e^{- \sqrt {C_{1} + \log {\left (t \right )}^{2} + 2 \log {\left (t \right )}} - 1}, \ r{\left (t \right )} = e^{\sqrt {C_{1} + \log {\left (t \right )}^{2} + 2 \log {\left (t \right )}} - 1}\right ] \]