74.14.11 problem 11

Internal problem ID [16416]
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 : 11
Date solved : Tuesday, January 28, 2025 at 09:07:41 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 78

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

\[ y = \frac {i \left ({\mathrm e}^{2 i t}-{\mathrm e}^{-2 i t}\right ) \ln \left (\frac {i {\mathrm e}^{2 i t}-1}{-{\mathrm e}^{2 i t}+i}\right )}{16}+\frac {c_{1} \sin \left (2 t \right )}{2}-\frac {c_{2} \cos \left (2 t \right )}{2}+\frac {\ln \left ({\mathrm e}^{i t}\right )}{4}-\frac {\ln \left ({\mathrm e}^{2 i t}-i\right )}{8}-\frac {\ln \left ({\mathrm e}^{2 i t}+i\right )}{8}+c_{3} \]

✓ Solution by Mathematica

Time used: 60.078 (sec). Leaf size: 54

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

\[ y(t)\to \int _1^t\left (-\frac {1}{4} \text {arctanh}(\sin (2 K[1])) \cos (2 K[1])+c_1 \cos (2 K[1])+\frac {1}{4} (4 c_2-1) \sin (2 K[1])\right )dK[1]+c_3 \]