76.17.9 problem 18

Internal problem ID [17695]
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 : 18
Date solved : Tuesday, January 28, 2025 at 10:59:59 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 169

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

\[ y = \frac {\sin \left (\frac {t}{2}\right ) \sqrt {2-\sqrt {2}}\, \left (1+\sqrt {2}\right ) \operatorname {arctanh}\left (\sin \left (\frac {t}{2}\right ) \sqrt {2-\sqrt {2}}\, \left (2+\sqrt {2}\right )\right )}{2}+\frac {\sin \left (\frac {t}{2}\right ) \sqrt {2+\sqrt {2}}\, \left (\sqrt {2}-1\right ) \operatorname {arctanh}\left (\sin \left (\frac {t}{2}\right ) \sqrt {2+\sqrt {2}}\, \left (\sqrt {2}-2\right )\right )}{2}+\frac {\cos \left (\frac {t}{2}\right ) \sqrt {2+\sqrt {2}}\, \left (\sqrt {2}-1\right ) \operatorname {arctanh}\left (\cos \left (\frac {t}{2}\right ) \sqrt {2+\sqrt {2}}\, \left (\sqrt {2}-2\right )\right )}{2}+\frac {\cos \left (\frac {t}{2}\right ) \sqrt {2-\sqrt {2}}\, \left (1+\sqrt {2}\right ) \operatorname {arctanh}\left (\cos \left (\frac {t}{2}\right ) \sqrt {2-\sqrt {2}}\, \left (2+\sqrt {2}\right )\right )}{2}+\cos \left (\frac {t}{2}\right ) c_{1} +\sin \left (\frac {t}{2}\right ) c_{2} \]

✓ Solution by Mathematica

Time used: 110.653 (sec). Leaf size: 5591

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

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