67.4.4 problem Problem 2(d)

Internal problem ID [14048]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 5.6 Laplace transform. Nonhomogeneous equations. Problems page 368
Problem number : Problem 2(d)
Date solved : Tuesday, January 28, 2025 at 06:13:13 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using Laplace method With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=2 \end{align*}

✓ Solution by Maple

Time used: 8.723 (sec). Leaf size: 15

dsolve([diff(y(t),t$2)-4*diff(y(t),t)+4*y(t)=t^2*exp(2*t),y(0) = 1, D(y)(0) = 2],y(t), singsol=all)
 

\[ y = \frac {{\mathrm e}^{2 t} \left (t^{4}+12\right )}{12} \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 19

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

\[ y(t)\to \frac {1}{12} e^{2 t} \left (t^4+12\right ) \]