28.4.43 problem 7.43

Internal problem ID [4575]
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 : 7.43
Date solved : Tuesday, March 04, 2025 at 06:52:40 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 )+{\mathrm e}^{2 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=-2 x_{1} \left (t \right )+3 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 47

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t)+exp(2*t), diff(x__2(t),t) = -2*x__1(t)+3*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 54

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

\begin{align*} \text {x1}(t)\to e^{2 t} (c_1 \cos (t)+(c_2-c_1) \sin (t)-1) \\ \text {x2}(t)\to e^{2 t} (c_2 \cos (t)+(c_2-2 c_1) \sin (t)-2) \\ \end{align*}

✓ Sympy. Time used: 0.168 (sec). Leaf size: 99

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

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