48.5.4 problem Problem 5.4

Internal problem ID [7569]
Book : THEORY OF DIFFERENTIAL EQUATIONS IN ENGINEERING AND MECHANICS. K.T. CHAU, CRC Press. Boca Raton, FL. 2018
Section : Chapter 5. Systems of First Order Differential Equations. Section 5.11 Problems. Page 360
Problem number : Problem 5.4
Date solved : Sunday, March 30, 2025 at 12:15:42 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{1} \left (t \right )+2 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.132 (sec). Leaf size: 57

ode:=[diff(x__1(t),t) = 4*x__1(t)-x__2(t), diff(x__2(t),t) = 5*x__1(t)+2*x__2(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 70

ode={D[ x1[t],t]==4*x1[t]-x2[t],D[ x2[t],t]==5*x1[t]+2*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{3 t} (2 c_1 \cos (2 t)+(c_1-c_2) \sin (2 t)) \\ \text {x2}(t)\to \frac {1}{2} e^{3 t} (2 c_2 \cos (2 t)+(5 c_1-c_2) \sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.149 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-4*x__1(t) + x__2(t) + Derivative(x__1(t), t),0),Eq(-5*x__1(t) - 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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