76.6.14 problem 16

Internal problem ID [17398]
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 : 16
Date solved : Monday, March 31, 2025 at 04:12:50 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 )-4\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-y \left (t \right )-6 \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 49

ode:=[diff(x(t),t) = -x(t)-4*y(t)-4, diff(y(t),t) = x(t)-y(t)-6]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= 4+{\mathrm e}^{-t} \left (\cos \left (2 t \right ) c_1 +\sin \left (2 t \right ) c_2 \right ) \\ y \left (t \right ) &= -2-\frac {{\mathrm e}^{-t} \left (\cos \left (2 t \right ) c_2 -\sin \left (2 t \right ) c_1 \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.177 (sec). Leaf size: 185

ode={D[x[t],t]==-x[t]-4*y[t]-4,D[y[t],t]==x[t]-y[t]-6}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-t} \left (\cos (2 t) \int _1^t-4 e^{K[1]} (\cos (2 K[1])+3 \sin (2 K[1]))dK[1]-2 \sin (2 t) \int _1^t-2 e^{K[2]} (3 \cos (2 K[2])-\sin (2 K[2]))dK[2]+c_1 \cos (2 t)-2 c_2 \sin (2 t)\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (2 \cos (2 t) \int _1^t-2 e^{K[2]} (3 \cos (2 K[2])-\sin (2 K[2]))dK[2]+\sin (2 t) \int _1^t-4 e^{K[1]} (\cos (2 K[1])+3 \sin (2 K[1]))dK[1]+2 c_2 \cos (2 t)+c_1 \sin (2 t)\right ) \\ \end{align*}

✓ Sympy. Time used: 0.232 (sec). Leaf size: 82

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) + 4*y(t) + Derivative(x(t), t) + 4,0),Eq(-x(t) + y(t) + Derivative(y(t), t) + 6,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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