28.4.45 problem 7.45

Internal problem ID [4577]
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 : 7.45
Date solved : Sunday, March 30, 2025 at 03:26:43 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )+27 t\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+4 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.127 (sec). Leaf size: 49

ode:=[diff(x__1(t),t) = 2*x__1(t)+x__2(t)+27*t, diff(x__2(t),t) = -x__1(t)+4*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{3 t} c_2 +{\mathrm e}^{3 t} t c_1 -12 t -5 \\ x_{2} \left (t \right ) &= {\mathrm e}^{3 t} c_2 +{\mathrm e}^{3 t} t c_1 +{\mathrm e}^{3 t} c_1 -2-3 t \\ \end{align*}

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 62

ode={D[x1[t],t]==2*x1[t]+x2[t]+27*t,D[x2[t],t]==-x1[t]+4*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 e^{3 t}+t \left (-12+(c_2-c_1) e^{3 t}\right )-5 \\ \text {x2}(t)\to c_2 e^{3 t}+t \left (-3+(c_2-c_1) e^{3 t}\right )-2 \\ \end{align*}

✓ Sympy. Time used: 0.178 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-27*t - 2*x__1(t) - x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) - 4*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} t e^{3 t} - 12 t + \left (C_{1} - C_{2}\right ) e^{3 t} - 5, \ x^{2}{\left (t \right )} = - C_{1} t e^{3 t} - C_{2} e^{3 t} - 3 t - 2\right ] \]