74.12.6 problem 6

Internal problem ID [16313]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.4, page 163
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 09:02:57 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 48

dsolve(diff(y(t),t$2)+1/4*y(t)=sec(t/2)+csc(t/2),y(t), singsol=all)
 

\[ y = 4 \cos \left (\frac {t}{2}\right ) \ln \left (\cos \left (\frac {t}{2}\right )\right )+4 \sin \left (\frac {t}{2}\right ) \ln \left (\sin \left (\frac {t}{2}\right )\right )+\left (-2 t +c_{1} \right ) \cos \left (\frac {t}{2}\right )+2 \sin \left (\frac {t}{2}\right ) \left (t +\frac {c_{2}}{2}\right ) \]

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 50

DSolve[D[y[t],{t,2}]+1/4*y[t]==Sec[t/2]+Csc[t/2],y[t],t,IncludeSingularSolutions -> True]
 

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