28.4.46 problem 7.46

Internal problem ID [4578]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.46
Date solved : Tuesday, March 04, 2025 at 06:52:44 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.042 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 51

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

\begin{align*} \text {x1}(t)\to e^t \left (t^2+t+2 c_1 t-c_2 t+c_1\right ) \\ \text {x2}(t)\to e^t \left (2 t^2+4 c_1 t-2 c_2 t+c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.134 (sec). Leaf size: 53

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

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