14.20.8 problem 8

Internal problem ID [2705]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 8
Date solved : Sunday, March 30, 2025 at 12:15:17 AM
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 )&=4 x \left (t \right )-y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 5 \end{align*}

✓ Maple. Time used: 0.172 (sec). Leaf size: 37

ode:=[diff(x(t),t) = 3*x(t)-2*y(t), diff(y(t),t) = 4*x(t)-y(t)]; 
ic:=x(0) = 1y(0) = 5; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{t} \left (-4 \sin \left (2 t \right )+\cos \left (2 t \right )\right ) \\ y \left (t \right ) &= {\mathrm e}^{t} \left (-3 \sin \left (2 t \right )+5 \cos \left (2 t \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 40

ode={D[x[t],t]==3*x[t]-2*y[t],D[y[t],t]==4*x[t]-y[t]}; 
ic={x[0]==1,y[0]==5}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^t (\cos (2 t)-4 \sin (2 t)) \\ y(t)\to e^t (5 \cos (2 t)-3 \sin (2 t)) \\ \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 54

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(-4*x(t) + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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