70.3.6 problem 6

Internal problem ID [18664]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.4 (Differences between linear and nonlinear equations). Problems at page 79
Problem number : 6
Date solved : Thursday, October 02, 2025 at 03:19:16 PM
CAS classification : [_linear]

\begin{align*} y+\ln \left (t \right ) y^{\prime }&=\cot \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y \left (2\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.231 (sec). Leaf size: 46

ode:=ln(t)*diff(y(t),t)+y(t) = cot(t); 
ic:=[y(2) = 3]; 
dsolve([ode,op(ic)],y(t), singsol=all);
 

\[ y = \left (\int _{2}^{t}\frac {\cot \left (\textit {\_z1} \right ) {\mathrm e}^{-\operatorname {Ei}_{1}\left (-\ln \left (\textit {\_z1} \right )\right )}}{\ln \left (\textit {\_z1} \right )}d \textit {\_z1} +3 \,{\mathrm e}^{-\operatorname {Ei}_{1}\left (-\ln \left (2\right )\right )}\right ) {\mathrm e}^{\operatorname {Ei}_{1}\left (-\ln \left (t \right )\right )} \]

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 59

ode=Log[t]*D[y[t],t]+y[t]==Cot[t]; 
ic={y[2]==3}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \exp \left (\int _2^t-\frac {1}{\log (K[1])}dK[1]\right ) \left (\int _2^t\frac {\exp \left (-\int _2^{K[2]}-\frac {1}{\log (K[1])}dK[1]\right ) \cot (K[2])}{\log (K[2])}dK[2]+3\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(y(t) + log(t)*Derivative(y(t), t) - 1/tan(t),0) 
ics = {y(2): 3} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out