13.10.6 problem 6

Internal problem ID [2407]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 2.4, The method of variation of parameters. Page 154
Problem number : 6
Date solved : Monday, January 27, 2025 at 05:51:53 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 13

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

\[ y = \frac {4 t^{{9}/{2}} {\mathrm e}^{-2 t}}{63} \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 19

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

\[ y(t)\to \frac {4}{63} e^{-2 t} t^{9/2} \]