57.15.7 problem 6(g)

Internal problem ID [14471]
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 : Thursday, October 02, 2025 at 09:37:35 AM
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: 0.529 (sec). Leaf size: 48

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,op(ic)],x(t),method='laplace');
 

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

✓ Mathematica. Time used: 0.028 (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]
 

\begin{align*} 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 ) \end{align*}

✓ Sympy. Time used: 1.682 (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)
 

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