73.22.18 problem 31.7 (k)

Internal problem ID [15645]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 31. Delta Functions. Additional Exercises. page 572
Problem number : 31.7 (k)
Date solved : Tuesday, January 28, 2025 at 08:03:33 AM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} y^{\prime \prime \prime }+9 y^{\prime }&=\delta \left (t -1\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 9.553 (sec). Leaf size: 18

dsolve([diff(y(t),t$3)+9*diff(y(t),t)=Dirac(t-1),y(0) = 0, D(y)(0) = 0, (D@@2)(y)(0) = 0],y(t), singsol=all)
 

\[ y = -\frac {\operatorname {Heaviside}\left (t -1\right ) \left (-1+\cos \left (3 t -3\right )\right )}{9} \]

✓ Solution by Mathematica

Time used: 60.035 (sec). Leaf size: 224

DSolve[{D[ y[t],{t,3}]+9*D[y[t],t]==DiracDelta[t-1],{y[0]==0,Derivative[1][y][0] ==0,Derivative[2][y][0] ==0}},y[t],t,IncludeSingularSolutions -> True]
 

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