63.15.7 problem 6(g)

Internal problem ID [13111]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 3, Laplace transform. Section 3.2.1 Initial value problems. Exercises page 156
Problem number : 6(g)
Date solved : Wednesday, March 05, 2025 at 09:17:30 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x^{\prime \prime }+\frac {2 x^{\prime }}{5}+2 x&=1-\operatorname {Heaviside}\left (t -5\right ) \end{align*}

Using Laplace method With initial conditions

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

✓ Maple. Time used: 13.320 (sec). Leaf size: 55

ode:=diff(diff(x(t),t),t)+2/5*diff(x(t),t)+2*x(t) = 1-Heaviside(t-5); 
ic:=x(0) = 0, D(x)(0) = 0; 
dsolve([ode,ic],x(t),method='laplace');
 

\[ x \left (t \right ) = \frac {1}{2}+\left (\frac {1}{4}+\frac {i}{28}\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{\left (-\frac {1}{5}-\frac {7 i}{5}\right ) \left (t -5\right )}+\left (\frac {1}{4}-\frac {i}{28}\right ) \operatorname {Heaviside}\left (t -5\right ) {\mathrm e}^{\left (-\frac {1}{5}+\frac {7 i}{5}\right ) \left (t -5\right )}+\frac {\left (-7 \cos \left (\frac {7 t}{5}\right )-\sin \left (\frac {7 t}{5}\right )\right ) {\mathrm e}^{-\frac {t}{5}}}{14}-\frac {\operatorname {Heaviside}\left (t -5\right )}{2} \]

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 91

ode=D[x[t],{t,2}]+4/10*D[x[t],t]+2*x[t]==1-UnitStep[t-5]; 
ic={x[0]==0,Derivative[1][x][0 ]==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\[ x(t)\to -\frac {1}{14} e^{-t/5} \left (-\theta (5-t) \left (7 e^{t/5}+e \sin \left (7-\frac {7 t}{5}\right )-7 e \cos \left (7-\frac {7 t}{5}\right )\right )+e \sin \left (7-\frac {7 t}{5}\right )+\sin \left (\frac {7 t}{5}\right )-7 e \cos \left (7-\frac {7 t}{5}\right )+7 \cos \left (\frac {7 t}{5}\right )\right ) \]

✓ Sympy. Time used: 2.761 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(2*x(t) + Heaviside(t - 5) + 2*Derivative(x(t), t)/5 + Derivative(x(t), (t, 2)) - 1,0) 
ics = {x(0): 0, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \left (- \frac {\sin {\left (\frac {7 t}{5} \right )}}{14} + \frac {e \sin {\left (\frac {7 t}{5} - 7 \right )} \theta \left (t - 5\right )}{14} - \frac {\cos {\left (\frac {7 t}{5} \right )}}{2} + \frac {e \cos {\left (\frac {7 t}{5} - 7 \right )} \theta \left (t - 5\right )}{2}\right ) e^{- \frac {t}{5}} - \frac {\theta \left (t - 5\right )}{2} + \frac {1}{2} \]