74.14.13 problem 13

Internal problem ID [16347]
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 : 13
Date solved : Monday, March 31, 2025 at 02:50:49 PM
CAS classification : [[_high_order, _missing_y]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 331

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

\[ y = \frac {\left ({\mathrm e}^{4 i t} \left (\operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}-1+2 i {\mathrm e}^{2 i t}}{{\mathrm e}^{4 i t}+1}\right ) \pi -\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) \pi +\operatorname {csgn}\left ({\mathrm e}^{4 i t}-1+2 i {\mathrm e}^{2 i t}\right ) \pi +\operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}-1+2 i {\mathrm e}^{2 i t}}{{\mathrm e}^{4 i t}+1}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) \operatorname {csgn}\left ({\mathrm e}^{4 i t}-1+2 i {\mathrm e}^{2 i t}\right ) \pi +2 i \ln \left (i \left ({\mathrm e}^{2 i t}+i\right )^{2}\right )-2 i \ln \left ({\mathrm e}^{4 i t}+1\right )+8 i c_2 -8 c_1 \right )+2 \left (-8 i t \ln \left ({\mathrm e}^{i t}\right )+32 c_3 t -8 t^{2}-\ln \left ({\mathrm e}^{4 i t}+1\right )-\ln \left ({\mathrm e}^{-4 i t}+1\right )+32 c_4 \right ) {\mathrm e}^{2 i t}-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) \operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right ) \operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )+\pi \,\operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right )+\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right )-\pi \,\operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )-2 i \ln \left (i \left (-{\mathrm e}^{-2 i t}+i\right )^{2}\right )+2 i \ln \left ({\mathrm e}^{-4 i t}+1\right )-8 i c_2 -8 c_1 \right ) {\mathrm e}^{-2 i t}}{64} \]

✓ Mathematica. Time used: 60.075 (sec). Leaf size: 62

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

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

✓ Sympy. Time used: 1.289 (sec). Leaf size: 148

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(4*Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)) - 1/cos(2*t)**2,0) 
ics = {} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = C_{1} + C_{2} t + C_{4} \cos {\left (2 t \right )} + \left (C_{3} + \frac {\log {\left (\sin {\left (2 t \right )} - 1 \right )}}{32} - \frac {\log {\left (\sin {\left (2 t \right )} + 1 \right )}}{32}\right ) \sin {\left (2 t \right )} + \frac {- \log {\left (\frac {2}{\cos {\left (2 t \right )} + 1} \right )} + \log {\left (\tan {\left (t \right )} - 1 \right )} + \log {\left (\tan {\left (t \right )} + 1 \right )}}{32 \cos {\left (2 t \right )}} + \frac {\log {\left (\frac {2}{\cos {\left (2 t \right )} + 1} \right )}}{32} - \frac {\log {\left (\tan {\left (t \right )} - 1 \right )}}{32} - \frac {\log {\left (\tan {\left (t \right )} + 1 \right )}}{32} + \frac {\log {\left (\tan {\left (t \right )} - 1 \right )}}{16 \left (\tan ^{2}{\left (t \right )} - 1\right )} + \frac {\log {\left (\tan {\left (t \right )} + 1 \right )}}{16 \left (\tan ^{2}{\left (t \right )} - 1\right )} - \frac {\log {\left (\frac {1}{\cos ^{2}{\left (t \right )}} \right )}}{16 \left (\tan ^{2}{\left (t \right )} - 1\right )} \]