76.3.6 problem 6

Internal problem ID [17377]
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 : Tuesday, January 28, 2025 at 10:03:35 AM
CAS classification : [_linear]

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

With initial conditions

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

✓ Solution by Maple

Time used: 2.920 (sec). Leaf size: 46

dsolve([ln(t)*diff(y(t),t)+y(t)=cot(t),y(2) = 3],y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.098 (sec). Leaf size: 59

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

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