74.14.14 problem 14

Internal problem ID [16340]
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 : 14
Date solved : Thursday, March 13, 2025 at 08:10:44 AM
CAS classification : [[_high_order, _missing_y]]

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

✓ Maple. Time used: 0.101 (sec). Leaf size: 310

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

\[ y = -\frac {\pi \,{\mathrm e}^{-2 i t} \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) \operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right )+1\right ) \operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )}{64}+\frac {{\mathrm e}^{2 i t} \pi \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) \operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right )+1\right ) \operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1}{{\mathrm e}^{4 i t}+1}\right )}{64}+\frac {\pi \,\operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right ) {\mathrm e}^{-2 i t}}{64}+\frac {\left (-1+i {\mathrm e}^{-2 i t}\right ) \ln \left ({\mathrm e}^{-4 i t}+1\right )}{32}+\frac {\pi \,\operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right ) {\mathrm e}^{2 i t}}{64}+\frac {\left (-i {\mathrm e}^{2 i t}-1\right ) \ln \left ({\mathrm e}^{4 i t}+1\right )}{32}+\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) {\mathrm e}^{-2 i t}}{64}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) {\mathrm e}^{2 i t}}{64}+\frac {i {\mathrm e}^{2 i t} \ln \left (i \left ({\mathrm e}^{2 i t}+i\right )^{2}\right )}{32}-\frac {i {\mathrm e}^{-2 i t} \ln \left (i \left (-{\mathrm e}^{-2 i t}+i\right )^{2}\right )}{32}-\frac {{\mathrm e}^{-2 i t} \left (c_{2} i+c_{1} \right )}{8}+\frac {\left (c_{2} i-c_{1} \right ) {\mathrm e}^{2 i t}}{8}-\frac {i t \ln \left ({\mathrm e}^{i t}\right )}{4}-\frac {3 t^{2}}{8}+c_{3} t +c_4 \]

✓ Mathematica. Time used: 60.146 (sec). Leaf size: 100

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

\[ y(t)\to \int _1^t\int _1^{K[3]}\frac {1}{8} \left (8 c_1 \cos (2 K[2])+8 \int _1^{K[2]}-\frac {1}{2} \sin (2 K[1]) \tan ^2(2 K[1])dK[1] \cos (2 K[2])+\cos (4 K[2])+2 \text {arctanh}(\sin (2 K[2])) \sin (2 K[2])+8 c_2 \sin (2 K[2])-1\right )dK[2]dK[3]+c_4 t+c_3 \]

✓ Sympy. Time used: 0.792 (sec). Leaf size: 68

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

\[ y{\left (t \right )} = C_{1} + C_{2} t + C_{4} \cos {\left (2 t \right )} - \frac {t^{2}}{8} + \left (C_{3} + \frac {\log {\left (\sin {\left (2 t \right )} - 1 \right )}}{32} - \frac {\log {\left (\sin {\left (2 t \right )} + 1 \right )}}{32}\right ) \sin {\left (2 t \right )} + \frac {\log {\left (\frac {1}{\left (1 - \cos {\left (2 t \right )}\right )^{2} + 2 \cos {\left (2 t \right )} - 1} \right )}}{32} \]