73.27.6 problem 38.6

Internal problem ID [15691]
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 : 38.6
Date solved : Monday, March 31, 2025 at 01:45:12 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = -21 \end{align*}

✓ Maple. Time used: 0.132 (sec). Leaf size: 33

ode:=[diff(x(t),t) = 4*x(t)+2*y(t), diff(y(t),t) = 3*x(t)-y(t)]; 
ic:=x(0) = 0y(0) = -21; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= 6 \,{\mathrm e}^{-2 t}-6 \,{\mathrm e}^{5 t} \\ y \left (t \right ) &= -18 \,{\mathrm e}^{-2 t}-3 \,{\mathrm e}^{5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 36

ode={D[x[t],t]==4*x[t]+2*y[t],D[y[t],t]==3*x[t]-y[t]}; 
ic={x[0]==0,y[0]==-21}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -6 e^{-2 t} \left (e^{7 t}-1\right ) \\ y(t)\to -3 e^{-2 t} \left (e^{7 t}+6\right ) \\ \end{align*}

✓ Sympy. Time used: 0.102 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) - 2*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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