68.14.14 problem 14
Internal
problem
ID
[17706]
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, October 02, 2025 at 02:27:13 PM
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: 0.003 (sec). Leaf size: 331
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);
\[
y = \frac {\left ({\mathrm e}^{4 i t} \left (\operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}-1+2 i {\mathrm e}^{2 i t}}{{\mathrm e}^{4 i t}+1}\right ) \pi -\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right )+\pi \,\operatorname {csgn}\left ({\mathrm e}^{4 i t}-1+2 i {\mathrm e}^{2 i t}\right )+\operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}-1+2 i {\mathrm e}^{2 i t}}{{\mathrm e}^{4 i t}+1}\right ) \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) \operatorname {csgn}\left ({\mathrm e}^{4 i t}-1+2 i {\mathrm e}^{2 i t}\right )+2 i \ln \left (i \left ({\mathrm e}^{2 i t}+i\right )^{2}\right )-2 i \ln \left ({\mathrm e}^{4 i t}+1\right )+8 i c_2 -8 c_1 \right )+2 \left (-8 i t \ln \left ({\mathrm e}^{i t}\right )+32 c_3 t -12 t^{2}-\ln \left ({\mathrm e}^{4 i t}+1\right )-\ln \left ({\mathrm e}^{-4 i t}+1\right )+32 c_4 \right ) {\mathrm e}^{2 i t}-\pi \,\operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )+\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right )+\pi \,\operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right )-\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) \operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right ) \operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )+2 i \ln \left ({\mathrm e}^{-4 i t}+1\right )-2 i \ln \left (i \left (-{\mathrm e}^{-2 i t}+i\right )^{2}\right )-8 i c_2 -8 c_1 \right ) {\mathrm e}^{-2 i t}}{64}
\]
✓ Mathematica. Time used: 60.092 (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]
\begin{align*} 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 \end{align*}
✓ Sympy. Time used: 0.534 (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)
\[
y{\left (t \right )} = C_{1} + C_{2} t + C_{4} \cos {\left (2 t \right )} - \frac {t^{2}}{8} + \left (C_{3} + \frac {\log {\left (\sin {\left (2 t \right )} - 1 \right )}}{32} - \frac {\log {\left (\sin {\left (2 t \right )} + 1 \right )}}{32}\right ) \sin {\left (2 t \right )} + \frac {\log {\left (\frac {1}{\left (1 - \cos {\left (2 t \right )}\right )^{2} + 2 \cos {\left (2 t \right )} - 1} \right )}}{32}
\]