14.26.5 problem 5

Internal problem ID [2778]
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 : 5
Date solved : Tuesday, March 04, 2025 at 02:41:26 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )-4 x_{2} \left (t \right )+{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=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 ) = 1\\ x_{2} \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 31

ode:=[diff(x__1(t),t) = 3*x__1(t)-4*x__2(t)+exp(t), diff(x__2(t),t) = x__1(t)-x__2(t)+exp(t)]; 
ic:=x__1(0) = 1x__2(0) = 1; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 31

ode={D[x1[t],t]==3*x1[t]-4*x2[t]+Exp[t],D[ x2[t],t]==1*x1[t]-1*x2[t]+Exp[t]}; 
ic={x1[0]==1,x2[0]==1}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.131 (sec). Leaf size: 51

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) + 4*x__2(t) - exp(t) + Derivative(x__1(t), t),0),Eq(-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)
 

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