76.17.8 problem 17

Internal problem ID [17694]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.7 (Variation of parameters). Problems at page 280
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 10:59:54 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+4 y&=2 \csc \left (\frac {t}{2}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 68

dsolve(diff(y(t),t$2)+4*y(t)=2*csc(t/2),y(t), singsol=all)
 

\[ y = 8 \sin \left (\frac {t}{2}\right ) \ln \left (\csc \left (\frac {t}{2}\right )-\cot \left (\frac {t}{2}\right )\right ) \left (2 \cos \left (\frac {t}{2}\right )^{3}-\cos \left (\frac {t}{2}\right )\right )+16 \sin \left (\frac {t}{2}\right ) \cos \left (\frac {t}{2}\right )^{2}+\cos \left (2 t \right ) c_{1} +\sin \left (2 t \right ) c_{2} -\frac {8 \sin \left (\frac {t}{2}\right )}{3} \]

✓ Solution by Mathematica

Time used: 0.103 (sec). Leaf size: 64

DSolve[D[y[t],{t,2}]+4*y[t]==2*Csc[t/2],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {4}{3} \sin \left (\frac {t}{2}\right )+4 \sin \left (\frac {3 t}{2}\right )+2 \sin (2 t) \log \left (\sin \left (\frac {t}{4}\right )\right )+c_1 \cos (2 t)+c_2 \sin (2 t)-2 \sin (2 t) \log \left (\cos \left (\frac {t}{4}\right )\right ) \]