64.20.12 problem 12

Internal problem ID [13655]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 9, The Laplace transform. Section 9.3, Exercises page 452
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 05:54:23 AM
CAS classification : [[_3rd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime \prime }-6 y^{\prime \prime }+11 y^{\prime }-6 y&=36 t \,{\mathrm e}^{4 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=0\\ y^{\prime \prime }\left (0\right )&=-6 \end{align*}

✓ Solution by Maple

Time used: 8.418 (sec). Leaf size: 25

dsolve([diff(y(t),t$3)-6*diff(y(t),t$2)+11*diff(y(t),t)-6*y(t)=36*t*exp(4*t),y(0) = -1, D(y)(0) = 0, (D@@2)(y)(0) = -6],y(t), singsol=all)
 

\[ y = -4 \,{\mathrm e}^{t}+14 \,{\mathrm e}^{3 t}+{\mathrm e}^{4 t} \left (6 t -11\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 29

DSolve[{D[y[t],{t,3}]-6*D[y[t],{t,2}]+11*D[y[t],t]-6*y[t]==36*t*Exp[4*t],{y[0]==-1,Derivative[1][y][0]==0,Derivative[2][y][0]==-6}},{y[t]},t,IncludeSingularSolutions -> True]
 

\[ y(t)\to e^t \left (e^{3 t} (6 t-11)+14 e^{2 t}-4\right ) \]