73.18.9 problem 27.1 (i)

Internal problem ID [15593]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 27. Differentiation and the Laplace transform. Additional Exercises. page 496
Problem number : 27.1 (i)
Date solved : Tuesday, January 28, 2025 at 08:02:56 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-5 y^{\prime }+6 y&=t^{2} {\mathrm e}^{4 t} \end{align*}

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=0\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 9.351 (sec). Leaf size: 27

dsolve([diff(y(t),t$2)-5*diff(y(t),t)+6*y(t)=t^2*exp(4*t),y(0) = 0, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = -\frac {7 \,{\mathrm e}^{2 t}}{4}+\frac {\left (2 t^{2}-6 t +7\right ) {\mathrm e}^{4 t}}{4} \]

✓ Solution by Mathematica

Time used: 0.022 (sec). Leaf size: 32

DSolve[{D[y[t],{t,2}]-5*D[y[t],t]+6*y[t]==t^2*Exp[4*t],{y[0]==0,Derivative[1][y][0] ==2}},y[t],t,IncludeSingularSolutions -> True]
 

\[ y(t)\to \frac {1}{4} e^{2 t} \left (e^{2 t} \left (2 t^2-6 t+7\right )-7\right ) \]