55.1.14 problem HW 5 problem 6

Internal problem ID [8710]
Book : Selected problems from homeworks from different courses
Section : Math 2520, summer 2021. Differential Equations and Linear Algebra. Normandale college, Bloomington, Minnesota
Problem number : HW 5 problem 6
Date solved : Sunday, March 30, 2025 at 01:24:19 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )+4 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 ) = 3\\ y \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.122 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 30

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

\begin{align*} x(t)\to e^{-5 t}+2 e^t \\ y(t)\to e^t-e^{-5 t} \\ \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) - 4*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)
 

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