14.26.4 problem 4

Internal problem ID [2777]
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 : 4
Date solved : Tuesday, March 04, 2025 at 02:41:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+2 \,{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )-{\mathrm e}^{t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 41

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t)+2*exp(t), diff(x__2(t),t) = 4*x__1(t)+x__2(t)-exp(t)]; 
ic:=x__1(0) = 0x__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {3 \,{\mathrm e}^{3 t}}{8}-\frac {5 \,{\mathrm e}^{-t}}{8}+\frac {{\mathrm e}^{t}}{4} \\ x_{2} \left (t \right ) &= \frac {3 \,{\mathrm e}^{3 t}}{4}+\frac {5 \,{\mathrm e}^{-t}}{4}-2 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 57

ode={D[x1[t],t]==1*x1[t]+1*x2[t]+2*Exp[t],D[ x2[t],t]==4*x1[t]+1*x2[t]-Exp[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}{8} e^{-t} \left (2 e^{2 t}+3 e^{4 t}-5\right ) \\ \text {x2}(t)\to \frac {5 e^{-t}}{4}-2 e^t+\frac {3 e^{3 t}}{4} \\ \end{align*}

✓ Sympy. Time used: 0.165 (sec). Leaf size: 41

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-x__1(t) - x__2(t) - 2*exp(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) - x__2(t) + exp(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1} e^{- t}}{2} + \frac {C_{2} e^{3 t}}{2} + \frac {e^{t}}{4}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{3 t} - 2 e^{t}\right ] \]