74.3.23 problem 18

Internal problem ID [15883]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.1, page 32
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 08:17:12 AM
CAS classification : [_linear]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.195 (sec). Leaf size: 52

dsolve([diff(y(t),t)+y(t)*sec(t)=t,y(0) = 0],y(t), singsol=all)
 

\[ y = \frac {i \pi ^{2}+12 i t^{2}-48 t \ln \left (1+i {\mathrm e}^{i t}\right )+48 i \operatorname {polylog}\left (2, -i {\mathrm e}^{i t}\right )-48 \operatorname {Catalan}}{24 \sec \left (t \right )+24 \tan \left (t \right )} \]

✓ Solution by Mathematica

Time used: 3.030 (sec). Leaf size: 813

DSolve[{D[y[t],t]+y[t]*Sec[t]==t,{y[0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {e^{-2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \left (-3 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \cos ^2\left (\frac {t}{2}\right ) t^3+\frac {3}{2} i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \sin (t) t^3-12 e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \cos ^2\left (\frac {t}{2}\right ) \log \left (2 \cos \left (\frac {t}{2}\right ) \left (\cos \left (\frac {t}{2}\right )-i \sin \left (\frac {t}{2}\right )\right )\right ) t^2+6 e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \log \left (2 \cos \left (\frac {t}{2}\right ) \left (\cos \left (\frac {t}{2}\right )-i \sin \left (\frac {t}{2}\right )\right )\right ) \sin (t) t^2-12 i \operatorname {PolyLog}\left (2,\frac {1}{2}+\frac {i}{2}\right ) \sin ^2\left (\frac {t}{2}\right ) t+12 i \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {i}{2}\right ) \sin ^2\left (\frac {t}{2}\right ) t+i \pi ^2 \sin ^2\left (\frac {t}{2}\right ) t+12 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \cos ^2\left (\frac {t}{2}\right ) \log \left (i \tan \left (\frac {t}{2}\right )+1\right ) \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (\tan \left (\frac {t}{2}\right )-1\right )\right ) t-12 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \cos ^2\left (\frac {t}{2}\right ) \log \left (1-i \tan \left (\frac {t}{2}\right )\right ) \log \left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (\tan \left (\frac {t}{2}\right )-1\right )\right ) t+12 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \cos ^2\left (\frac {t}{2}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i) \tan \left (\frac {t}{2}\right )+(1-i)\right )\right ) t-6 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \operatorname {PolyLog}(2,i \sin (t)-\cos (t)) (\cos (t)-\sin (t)+1) t-6 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1-i) \tan \left (\frac {t}{2}\right )+(1+i)\right )\right ) (\cos (t)-\sin (t)+1) t-6 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \log \left (i \tan \left (\frac {t}{2}\right )+1\right ) \log \left (\left (-\frac {1}{2}-\frac {i}{2}\right ) \left (\tan \left (\frac {t}{2}\right )-1\right )\right ) \sin (t) t+6 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \log \left (1-i \tan \left (\frac {t}{2}\right )\right ) \log \left (\left (-\frac {1}{2}+\frac {i}{2}\right ) \left (\tan \left (\frac {t}{2}\right )-1\right )\right ) \sin (t) t-6 i e^{2 \text {arctanh}\left (\tan \left (\frac {t}{2}\right )\right )} \operatorname {PolyLog}\left (2,\frac {1}{2} \left ((1+i) \tan \left (\frac {t}{2}\right )+(1-i)\right )\right ) \sin (t) t+6 i \operatorname {PolyLog}\left (2,\frac {1}{2}+\frac {i}{2}\right ) \sin (t) t-6 i \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {i}{2}\right ) \sin (t) t-\frac {1}{2} i \pi ^2 \sin (t) t+\pi ^2 \log \left (1-i \tan \left (\frac {t}{2}\right )\right )-\pi ^2 \log \left (i \tan \left (\frac {t}{2}\right )+1\right )-12 \log \left (1-i \tan \left (\frac {t}{2}\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{2}+\frac {i}{2}\right )+12 \log \left (i \tan \left (\frac {t}{2}\right )+1\right ) \operatorname {PolyLog}\left (2,\frac {1}{2}+\frac {i}{2}\right )+12 \log \left (1-i \tan \left (\frac {t}{2}\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {i}{2}\right )-12 \log \left (i \tan \left (\frac {t}{2}\right )+1\right ) \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {i}{2}\right )\right )}{3 \left (\cos (t) t+\sin (t) t-t+2 i \log \left (1-i \tan \left (\frac {t}{2}\right )\right )-2 i \log \left (i \tan \left (\frac {t}{2}\right )+1\right )\right )} \]