74.14.25 problem 25

Internal problem ID [16430]
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 : Tuesday, January 28, 2025 at 09:07:52 AM
CAS classification : [[_high_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 68

dsolve(diff(y(t),t$4)+diff(y(t),t$2)=tan(t)^2,y(t), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 60.095 (sec). Leaf size: 77

DSolve[D[y[t],{t,4}]+D[y[t],{t,2}]==Tan[t]^2,y[t],t,IncludeSingularSolutions -> True]
 

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