74.14.30 problem 30

Internal problem ID [16435]
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 : 30
Date solved : Tuesday, January 28, 2025 at 09:07:56 AM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime \prime }+y^{\prime \prime }&=t \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: 20

dsolve([diff(y(t),t$4)+diff(y(t),t$2)=t,y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 1, (D@@3)(y)(0) = 0],y(t), singsol=all)
 
\[ y = \frac {t^{3}}{6}-\cos \left (t \right )+\sin \left (t \right )-t +1 \]

Solution by Mathematica

Time used: 60.024 (sec). Leaf size: 81

DSolve[{D[y[t],{t,4}]+D[y[t],{t,2}]==t,{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==1,Derivative[3][y][0]==0}},y[t],t,IncludeSingularSolutions -> True]
 
\[ y(t)\to -t \int _1^0(\cos (K[1])+K[1]-\sin (K[1]))dK[1]+\int _1^t\int _1^{K[2]}(\cos (K[1])+K[1]-\sin (K[1]))dK[1]dK[2]-\int _1^0\int _1^{K[2]}(\cos (K[1])+K[1]-\sin (K[1]))dK[1]dK[2] \]