74.14.14 problem 14
Internal
problem
ID
[16419]
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
:
Tuesday, January 28, 2025 at 09:07:44 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*}
✓ Solution by Maple
Time used: 0.101 (sec). Leaf size: 310
dsolve(diff(y(t),t$4)+4*diff(y(t),t$2)=tan(2*t)^2,y(t), singsol=all)
\[
y = -\frac {\pi \,{\mathrm e}^{-2 i t} \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) \operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right )+1\right ) \operatorname {csgn}\left (\frac {2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1}{{\mathrm e}^{-4 i t}+1}\right )}{64}+\frac {{\mathrm e}^{2 i t} \pi \left (\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) \operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right )+1\right ) \operatorname {csgn}\left (\frac {{\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1}{{\mathrm e}^{4 i t}+1}\right )}{64}+\frac {\pi \,\operatorname {csgn}\left (2 i {\mathrm e}^{-2 i t}-{\mathrm e}^{-4 i t}+1\right ) {\mathrm e}^{-2 i t}}{64}+\frac {\left (-1+i {\mathrm e}^{-2 i t}\right ) \ln \left ({\mathrm e}^{-4 i t}+1\right )}{32}+\frac {\pi \,\operatorname {csgn}\left ({\mathrm e}^{4 i t}+2 i {\mathrm e}^{2 i t}-1\right ) {\mathrm e}^{2 i t}}{64}+\frac {\left (-i {\mathrm e}^{2 i t}-1\right ) \ln \left ({\mathrm e}^{4 i t}+1\right )}{32}+\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{-4 i t}+1}\right ) {\mathrm e}^{-2 i t}}{64}-\frac {\pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{4 i t}+1}\right ) {\mathrm e}^{2 i t}}{64}+\frac {i {\mathrm e}^{2 i t} \ln \left (i \left ({\mathrm e}^{2 i t}+i\right )^{2}\right )}{32}-\frac {i {\mathrm e}^{-2 i t} \ln \left (i \left (-{\mathrm e}^{-2 i t}+i\right )^{2}\right )}{32}-\frac {{\mathrm e}^{-2 i t} \left (c_{2} i+c_{1} \right )}{8}+\frac {\left (c_{2} i-c_{1} \right ) {\mathrm e}^{2 i t}}{8}-\frac {i t \ln \left ({\mathrm e}^{i t}\right )}{4}-\frac {3 t^{2}}{8}+c_{3} t +c_4
\]
✓ Solution by Mathematica
Time used: 60.146 (sec). Leaf size: 100
DSolve[D[y[t],{t,4}]+4*D[y[t],{t,2}]==Tan[2*t]^2,y[t],t,IncludeSingularSolutions -> True]
\[
y(t)\to \int _1^t\int _1^{K[3]}\frac {1}{8} \left (8 c_1 \cos (2 K[2])+8 \int _1^{K[2]}-\frac {1}{2} \sin (2 K[1]) \tan ^2(2 K[1])dK[1] \cos (2 K[2])+\cos (4 K[2])+2 \text {arctanh}(\sin (2 K[2])) \sin (2 K[2])+8 c_2 \sin (2 K[2])-1\right )dK[2]dK[3]+c_4 t+c_3
\]