74.12.31 problem 31

Internal problem ID [16338]
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 : 31
Date solved : Tuesday, January 28, 2025 at 09:05:12 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 55

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

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

✓ Solution by Mathematica

Time used: 0.342 (sec). Leaf size: 73

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

\[ y(t)\to \cos (t) \int _1^t-2 \left (\cos \left (\frac {K[1]}{2}\right )+\sin \left (\frac {K[1]}{2}\right )\right )dK[1]+\sin (t) \int _1^t2 \cot (K[2]) \left (\cos \left (\frac {K[2]}{2}\right )+\sin \left (\frac {K[2]}{2}\right )\right )dK[2]+c_1 \cos (t)+c_2 \sin (t) \]