74.14.27 problem 27

Internal problem ID [16432]
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 : 27
Date solved : Tuesday, January 28, 2025 at 09:07:53 AM
CAS classification : [[_high_order, _missing_y]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=1\\ y^{\prime \prime }\left (0\right )&=0\\ y^{\prime \prime \prime }\left (0\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 36.748 (sec). Leaf size: 63

dsolve([diff(y(t),t$4)+diff(y(t),t$2)=sec(t)^2,y(0) = 0, D(y)(0) = 1, (D@@2)(y)(0) = 0, (D@@3)(y)(0) = 0],y(t), singsol=all)
 

\[ y = \frac {\left (\int _{0}^{t}\left (\left (-{\mathrm e}^{-i \textit {\_z1}}-{\mathrm e}^{i \textit {\_z1}}\right ) \ln \left (\frac {i {\mathrm e}^{i \textit {\_z1}}-1}{-{\mathrm e}^{i \textit {\_z1}}+i}\right )-2 i \ln \left ({\mathrm e}^{i \textit {\_z1}}\right )-2 \textit {\_z1} +2 \sin \left (\textit {\_z1} \right )\right )d \textit {\_z1} \right )}{2}+t \]

✓ Solution by Mathematica

Time used: 60.050 (sec). Leaf size: 103

DSolve[{D[y[t],{t,4}]+D[y[t],{t,2}]==Sec[t]^2,{y[0]==0,Derivative[1][y][0] ==1,Derivative[2][y][0] ==0,Derivative[3][y][0]==0}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to t \left (-\int _1^0\left (\cos (K[1])+2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right ) \sin (K[1])-1\right )dK[1]\right )+\int _1^t\int _1^{K[2]}\left (\cos (K[1])+2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right ) \sin (K[1])-1\right )dK[1]dK[2]-\int _1^0\int _1^{K[2]}\left (\cos (K[1])+2 \text {arctanh}\left (\tan \left (\frac {K[1]}{2}\right )\right ) \sin (K[1])-1\right )dK[1]dK[2]+t \]