68.14.11 problem 11

Internal problem ID [17703]
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 : Thursday, October 02, 2025 at 02:27:11 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 88

ode:=diff(diff(diff(y(t),t),t),t)+4*diff(y(t),t) = tan(2*t); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {i \left ({\mathrm e}^{2 i t}-{\mathrm e}^{-2 i t}\right ) \ln \left (\frac {i {\mathrm e}^{2 i t}-1}{-{\mathrm e}^{2 i t}+i}\right )}{16}-\frac {\ln \left ({\mathrm e}^{2 i t}-i\right )}{8}-\frac {\ln \left ({\mathrm e}^{2 i t}+i\right )}{8}+\frac {\left (i c_1 -c_2 \right ) {\mathrm e}^{-2 i t}}{4}-\frac {{\mathrm e}^{2 i t} \left (i c_1 +c_2 \right )}{4}+c_3 +\frac {\ln \left ({\mathrm e}^{i t}\right )}{4} \]

✓ Mathematica. Time used: 60.052 (sec). Leaf size: 54

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

\begin{align*} 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 \end{align*}

✓ Sympy. Time used: 0.358 (sec). Leaf size: 46

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-tan(2*t) + 4*Derivative(y(t), t) + Derivative(y(t), (t, 3)),0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} + C_{3} \cos {\left (2 t \right )} + \left (C_{2} + \frac {\log {\left (\sin {\left (2 t \right )} - 1 \right )}}{16} - \frac {\log {\left (\sin {\left (2 t \right )} + 1 \right )}}{16}\right ) \sin {\left (2 t \right )} - \frac {\log {\left (\cos {\left (2 t \right )} \right )}}{8} \]