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]]
\[ \boxed {y^{\prime \prime }+y-2 \sec \left (\frac {t}{2}\right )=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 43
dsolve(diff(y(t),t$2)+y(t) = 2*sec(t/2),y(t), singsol=all)
\[ y \left (t \right ) = c_{2} \sin \left (t \right )+\cos \left (t \right ) c_{1} +\cos \left (\frac {t}{2}\right ) \left (-8 \left (\ln \left (2\right )+\ln \left (\csc \left (t \right ) \sin \left (\frac {t}{2}\right ) \left (\sin \left (\frac {t}{2}\right )+1\right )\right )\right ) \sin \left (\frac {t}{2}\right )+8\right ) \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 34
DSolve[y''[t]+y[t]== 2*Sec[t/2],y[t],t,IncludeSingularSolutions -> True]
\begin{align*} y(t)\to \sin (t) \left (-4 \text {arctanh}\left (\sin \left (\frac {t}{2}\right )\right )+c_2\right )+8 \cos \left (\frac {t}{2}\right )+c_1 \cos (t) \\ \end{align*}