74.14.29 problem 29

Internal problem ID [16434]
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 : 29
Date solved : Tuesday, January 28, 2025 at 09:07:55 AM
CAS classification : [[_high_order, _missing_y]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 15

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

\[ y = -2 \cos \left (t \right )-\frac {t \sin \left (t \right )}{2}+2 \]

✓ Solution by Mathematica

Time used: 60.046 (sec). Leaf size: 192

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

\[ y(t)\to -t \int _1^0\frac {1}{2} \left (\cos ^3(K[2])+\cos (K[2])-2 \sin (K[2]) \int _1^0\cos ^2(K[1])dK[1]+2 \sin (K[2]) \int _1^{K[2]}\cos ^2(K[1])dK[1]\right )dK[2]+\int _1^t\int _1^{K[3]}\frac {1}{2} \left (\cos ^3(K[2])+\cos (K[2])-2 \sin (K[2]) \int _1^0\cos ^2(K[1])dK[1]+2 \sin (K[2]) \int _1^{K[2]}\cos ^2(K[1])dK[1]\right )dK[2]dK[3]-\int _1^0\int _1^{K[3]}\frac {1}{2} \left (\cos ^3(K[2])+\cos (K[2])-2 \sin (K[2]) \int _1^0\cos ^2(K[1])dK[1]+2 \sin (K[2]) \int _1^{K[2]}\cos ^2(K[1])dK[1]\right )dK[2]dK[3] \]