10.9 problem 9

Internal problem ID [691]

Book: Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section: Chapter 3, Second order linear equations, section 3.6, Variation of Parameters. page 190
Problem number: 9.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]

Solve \begin {gather*} \boxed {y^{\prime \prime }+y-2 \sec \left (\frac {t}{2}\right )=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 47

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

\[ y \relax (t ) = \sin \relax (t ) c_{2}+\cos \relax (t ) c_{1}-4 \sin \relax (t ) \ln \left (\frac {1+\sin \left (\frac {t}{2}\right )}{\cos \left (\frac {t}{2}\right )}\right )+8 \sin \relax (t ) \sin \left (\frac {t}{2}\right )+8 \cos \left (\frac {t}{2}\right ) \cos \relax (t ) \]

Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 34

DSolve[y''[t]+y[t]== 2*Sec[t/2],y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to 8 \cos \left (\frac {t}{2}\right )+c_1 \cos (t)+\sin (t) \left (-4 \tanh ^{-1}\left (\sin \left (\frac {t}{2}\right )\right )+c_2\right ) \\ \end{align*}