74.14.12 problem 12

Internal problem ID [16417]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.6, page 187
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 09:07:42 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 41

dsolve(diff(y(t),t$3)+4*diff(y(t),t)=sec(2*t)*tan(2*t),y(t), singsol=all)
 

\[ y = -\frac {\ln \left (\sec \left (2 t \right )\right ) \cos \left (2 t \right )}{8}+\frac {\left (1-4 c_{2} \right ) \cos \left (2 t \right )}{8}+\frac {\left (4 c_{1} +2 t \right ) \sin \left (2 t \right )}{8}+c_{3} \]

✓ Solution by Mathematica

Time used: 2.150 (sec). Leaf size: 84

DSolve[D[ y[t],{t,3}]+4*D[y[t],t]==Sec[2*t]*Tan[2*t],y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{16} \left (\frac {\arctan (\tan (2 t)) (2 t \sin (2 t)+\cos (2 t))}{t}+(4-16 c_2) \cos ^2(t)+2 \cos (2 t) \log (\cos (2 t))+8 c_1 \sin (2 t)+\frac {4 \cos (2 t) \log (\cos (2 t))}{\log \left (\sec ^2(2 t)\right )}\right )+c_3 \]