74.14.25 problem 25

Internal problem ID [16351]
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, March 13, 2025 at 08:10:55 AM
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: 0.005 (sec). Leaf size: 68

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);
 

\[ y = \frac {\left (\int \left (\left (-{\mathrm e}^{-i t}-{\mathrm e}^{i t}\right ) \ln \left (\frac {i {\mathrm e}^{i t}-1}{-{\mathrm e}^{i t}+i}\right )-2 i \ln \left ({\mathrm e}^{i t}\right )+2 c_{1} \sin \left (t \right )-2 c_{2} \cos \left (t \right )-4 t \right )d t \right )}{2}+c_{3} t +c_4 \]

✓ Mathematica. Time used: 60.095 (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]
 

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

✓ Sympy. Time used: 0.482 (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)
 

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