76.6.8 problem 8

Internal problem ID [17392]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 8
Date solved : Monday, March 31, 2025 at 04:12:41 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.127 (sec). Leaf size: 45

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{3 t} \left (c_2 \cos \left (2 t \right )+c_1 \sin \left (2 t \right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{3 t} \left (\cos \left (2 t \right ) c_1 -\sin \left (2 t \right ) c_2 \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.069 (sec). Leaf size: 157

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

\begin{align*} x(t)\to e^{3 t} \left (-2 \sin (2 t) \int _1^t-2 e^{-3 K[2]} \sin (2 K[2])dK[2]+\cos (2 t) \int _1^t4 e^{-3 K[1]} \cos (2 K[1])dK[1]+c_1 \cos (2 t)-2 c_2 \sin (2 t)\right ) \\ y(t)\to \frac {1}{2} e^{3 t} \left (2 \cos (2 t) \int _1^t-2 e^{-3 K[2]} \sin (2 K[2])dK[2]+\sin (2 t) \int _1^t4 e^{-3 K[1]} \cos (2 K[1])dK[1]+2 c_2 \cos (2 t)+c_1 \sin (2 t)\right ) \\ \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 56

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

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