67.27.12 problem 38.10 (f)

Internal problem ID [17055]
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 (f)
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 )&=3 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+3 y \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.140 (sec). Leaf size: 44

ode:=[diff(x(t),t) = 3*x(t)+2*y(t), diff(y(t),t) = -2*x(t)+3*y(t)]; 
ic:=[x(0) = a__1, y(0) = a__2]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 47

ode={D[x[t],t]==3*x[t]+2*y[t],D[y[t],t]==-2*x[t]+3*y[t]}; 
ic={x[0]==a1,y[0]==a2}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{3 t} (\text {a1} \cos (2 t)+\text {a2} \sin (2 t))\\ y(t)&\to e^{3 t} (\text {a2} \cos (2 t)-\text {a1} \sin (2 t)) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 51

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

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