14.26.10 problem 10

Internal problem ID [2783]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 3. Systems of differential equations. Section 3.13 (Solving systems by Laplace transform). Page 370
Problem number : 10
Date solved : Tuesday, March 04, 2025 at 02:42:35 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )+1-\operatorname {Heaviside}\left (t -\pi \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.034 (sec). Leaf size: 107

ode:=[diff(x__1(t),t) = 3*x__1(t)-2*x__2(t)+1-Heaviside(t-Pi), diff(x__2(t),t) = 2*x__1(t)-2*x__2(t)]; 
ic:=x__1(0) = 0x__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {{\mathrm e}^{-t}}{3}+\frac {2 \,{\mathrm e}^{2 t}}{3}+\operatorname {Heaviside}\left (t -\pi \right )-\frac {\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-t +\pi }}{3}-1-\frac {2 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 t -2 \pi }}{3} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{2 t}}{3}+\frac {2 \,{\mathrm e}^{-t}}{3}+\operatorname {Heaviside}\left (t -\pi \right )-\frac {\operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{2 t -2 \pi }}{3}-1-\frac {2 \operatorname {Heaviside}\left (t -\pi \right ) {\mathrm e}^{-t +\pi }}{3} \\ \end{align*}

✓ Mathematica. Time used: 15.993 (sec). Leaf size: 140

ode={D[x1[t],t]==3*x1[t]-2*x2[t]+1-HeavisideTheta[t-Pi],D[x2[t],t]==2*x1[t]-2*x2[t]}; 
ic={x1[0]==0,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{3} e^{-t-2 \pi } \left (e^{2 \pi } \left (-3 e^t+2 e^{3 t}+1\right )-\left (2 e^{3 t}-3 e^{t+2 \pi }+e^{3 \pi }\right ) \theta (t-1) \theta (t-\pi )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-t-2 \pi } \left (e^{2 \pi } \left (e^t-1\right )^2 \left (e^t+2\right )-\left (e^{\pi }-e^t\right )^2 \left (e^t+2 e^{\pi }\right ) \theta (t-1) \theta (t-\pi )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.712 (sec). Leaf size: 119

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) + 2*x__2(t) + Heaviside(t - pi) + Derivative(x__1(t), t) - 1,0),Eq(-2*x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- t}}{2} + 2 C_{2} e^{2 t} - \frac {2 e^{2 t} \theta \left (t - \pi \right )}{3 e^{2 \pi }} + \theta \left (t - \pi \right ) - 1 - \frac {e^{\pi } e^{- t} \theta \left (t - \pi \right )}{3}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{2 t} - \frac {e^{2 t} \theta \left (t - \pi \right )}{3 e^{2 \pi }} + \theta \left (t - \pi \right ) - 1 - \frac {2 e^{\pi } e^{- t} \theta \left (t - \pi \right )}{3}\right ] \]