46.11.2 problem 2

Internal problem ID [9720]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.3. Page 354
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 06:32:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )+9 y \left (t \right )+2\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+11 y \left (t \right )+6 \end{align*}

✓ Maple. Time used: 0.130 (sec). Leaf size: 36

ode:=[diff(x(t),t) = 5*x(t)+9*y(t)+2, diff(y(t),t) = -x(t)+11*y(t)+6]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {1}{2}+{\mathrm e}^{8 t} \left (t c_1 +c_2 \right ) \\ y \left (t \right ) &= -\frac {1}{2}+\frac {{\mathrm e}^{8 t} \left (3 t c_1 +c_1 +3 c_2 \right )}{9} \\ \end{align*}

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 144

ode={D[x[t],t]==5*x[t]+9*y[t]+2,D[y[t],t]==-x[t]+11*y[t]+6}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{8 t} \left ((1-3 t) \int _1^te^{-8 K[1]} (2-48 K[1])dK[1]+9 t \int _1^te^{-8 K[2]} (6-16 K[2])dK[2]-3 c_1 t+9 c_2 t+c_1\right )\\ y(t)&\to e^{8 t} \left (-t \int _1^te^{-8 K[1]} (2-48 K[1])dK[1]+(3 t+1) \int _1^te^{-8 K[2]} (6-16 K[2])dK[2]+c_1 (-t)+3 c_2 t+c_2\right ) \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) - 9*y(t) + Derivative(x(t), t) - 2,0),Eq(x(t) - 11*y(t) + Derivative(y(t), t) - 6,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - 3 C_{1} t e^{8 t} + \left (C_{1} - 3 C_{2}\right ) e^{8 t} + \frac {1}{2}, \ y{\left (t \right )} = - C_{1} t e^{8 t} - C_{2} e^{8 t} - \frac {1}{2}\right ] \]