68.14.25 problem 25
Internal
problem
ID
[17717]
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
:
25
Date
solved
:
Thursday, October 02, 2025 at 02:27:18 PM
CAS
classification
:
[[_high_order, _missing_y]]
\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }&=\tan \left (t \right )^{2} \end{align*}
✓ Maple. Time used: 0.004 (sec). Leaf size: 201
ode:=diff(diff(diff(diff(y(t),t),t),t),t)+diff(diff(y(t),t),t) = tan(t)^2;
dsolve(ode,y(t), singsol=all);
\[
y = \left (\frac {\pi \left (-1+{\mathrm e}^{2 i t}\right ) \left (\operatorname {csgn}\left ({\mathrm e}^{2 i t}-1+2 i {\mathrm e}^{i t}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i t}+1}\right )+1\right ) \operatorname {csgn}\left (\frac {{\mathrm e}^{2 i t}-1+2 i {\mathrm e}^{i t}}{{\mathrm e}^{2 i t}+1}\right )}{4}+\frac {\pi \left (-1+{\mathrm e}^{2 i t}\right ) \operatorname {csgn}\left ({\mathrm e}^{2 i t}-1+2 i {\mathrm e}^{i t}\right )}{4}-\frac {\pi \left (-1+{\mathrm e}^{2 i t}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i t}+1}\right )}{4}-\frac {i \left (-{\mathrm e}^{2 i t}+1\right ) \ln \left (i \left ({\mathrm e}^{i t}+i\right )^{2}\right )}{2}+\left (-\frac {i {\mathrm e}^{2 i t}}{2}+\frac {i}{2}-{\mathrm e}^{i t}\right ) \ln \left ({\mathrm e}^{2 i t}+1\right )+\frac {\left (i c_2 -c_1 \right ) {\mathrm e}^{2 i t}}{2}-i t \ln \left ({\mathrm e}^{i t}\right ) {\mathrm e}^{i t}+{\mathrm e}^{i t} \left (-\frac {3 t^{2}}{2}+t \left (i+c_3 \right )+c_4 \right )-\frac {i c_2}{2}-\frac {c_1}{2}\right ) {\mathrm e}^{-i t}
\]
✓ Mathematica. Time used: 60.054 (sec). Leaf size: 77
ode=D[y[t],{t,4}]+D[y[t],{t,2}]==Tan[t]^2;
ic={};
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
\begin{align*} y(t)&\to \int _1^t\int _1^{K[3]}\left (-\sin ^2(K[2])+\text {arctanh}(\sin (K[2])) \sin (K[2])+c_2 \sin (K[2])+c_1 \cos (K[2])+\cos (K[2]) \int _1^{K[2]}-\sin (K[1]) \tan ^2(K[1])dK[1]\right )dK[2]dK[3]+c_4 t+c_3 \end{align*}
✓ Sympy. Time used: 0.320 (sec). Leaf size: 49
from sympy import *
t = symbols("t")
y = Function("y")
ode = Eq(-tan(t)**2 + Derivative(y(t), (t, 2)) + Derivative(y(t), (t, 4)),0)
ics = {}
dsolve(ode,func=y(t),ics=ics)
\[
y{\left (t \right )} = C_{1} + C_{2} t + C_{4} \cos {\left (t \right )} - \frac {t^{2}}{2} + \left (C_{3} + \frac {\log {\left (\sin {\left (t \right )} - 1 \right )}}{2} - \frac {\log {\left (\sin {\left (t \right )} + 1 \right )}}{2}\right ) \sin {\left (t \right )} + \frac {\log {\left (\frac {1}{\cos ^{2}{\left (t \right )}} \right )}}{2}
\]