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 : Tuesday, March 04, 2025 at 11:26:33 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*}

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

ode:=diff(diff(x(t),t),t)+4*diff(x(t),t)+13*x(t) = t*exp(-t); 
ic:=x(0) = 0, D(x)(0) = 2; 
dsolve([ode,ic],x(t),method='laplace');
 

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

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

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

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

✓ Sympy. Time used: 0.335 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-t*exp(-t) + 13*x(t) + 4*Derivative(x(t), t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 2} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (\frac {t}{10} + \left (\frac {16 \sin {\left (3 t \right )}}{25} + \frac {\cos {\left (3 t \right )}}{50}\right ) e^{- t} - \frac {1}{50}\right ) e^{- t} \]