67.27.11 problem 38.10 (e)

Internal problem ID [17054]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.10 (e)
Date solved : Thursday, October 02, 2025 at 01:42:23 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-13 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=2 \\ y \left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.141 (sec). Leaf size: 34

ode:=[diff(x(t),t) = 4*x(t)-13*y(t), diff(y(t),t) = x(t)]; 
ic:=[x(0) = 2, y(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{2 t} \left (-3 \sin \left (3 t \right )+2 \cos \left (3 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{2 t} \cos \left (3 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 37

ode={D[x[t],t]==4*x[t]-13*y[t],D[y[t],t]==x[t]}; 
ic={x[0]==2,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{2 t} (2 \cos (3 t)-3 \sin (3 t))\\ y(t)&\to e^{2 t} \cos (3 t) \end{align*}

✓ Sympy. Time used: 0.069 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 13*y(t) + Derivative(x(t), t),0),Eq(-x(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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