7.19.11 problem 37

Internal problem ID [551]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 4. Laplace transform methods. Section 4.3 (Translation and partial fractions). Problems at page 296
Problem number : 37
Date solved : Wednesday, February 05, 2025 at 03:45:46 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+4 x^{\prime }+13 x&=t \,{\mathrm e}^{-t} \end{align*}

Using Laplace method With initial conditions

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

✓ Solution by Maple

Time used: 0.224 (sec). Leaf size: 33

dsolve([diff(x(t),t$2)+4*diff(x(t),t)+13*x(t)=t*exp(-t),x(0) = 0, D(x)(0) = 2],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {{\mathrm e}^{-2 t} \left (\cos \left (3 t \right )+32 \sin \left (3 t \right )\right )}{50}+\frac {\left (-1+5 t \right ) {\mathrm e}^{-t}}{50} \]

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 18

DSolve[{D[x[t],{t,2}]+4*D[x[t],t]+13*x[t]==0,{x[0]==0,Derivative[1][x][0] ==2}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {2}{3} e^{-2 t} \sin (3 t) \]