63.20.6 problem 5

Internal problem ID [13149]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 218
Problem number : 5
Date solved : Wednesday, March 05, 2025 at 09:18:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=5 x \left (t \right )-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 ) = 2\\ y \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 33

ode:=[diff(x(t),t) = 5*x(t)-y(t), diff(y(t),t) = 3*x(t)+y(t)]; 
ic:=x(0) = 2y(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= -\frac {3 \,{\mathrm e}^{2 t}}{2}+\frac {7 \,{\mathrm e}^{4 t}}{2} \\ y &= -\frac {9 \,{\mathrm e}^{2 t}}{2}+\frac {7 \,{\mathrm e}^{4 t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 44

ode={D[x[t],t]==5*x[t]-y[t],D[y[t],t]==3*x[t]+y[t]}; 
ic={x[0]==2,y[0]==-1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{2 t} \left (7 e^{2 t}-3\right ) \\ y(t)\to \frac {1}{2} e^{2 t} \left (7 e^{2 t}-9\right ) \\ \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-5*x(t) + 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)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} e^{2 t}}{3} + C_{2} e^{4 t}, \ y{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{4 t}\right ] \]