76.6.4 problem 4

Internal problem ID [17388]
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 : 4
Date solved : Monday, March 31, 2025 at 04:12:36 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )+2 y \left (t \right )+4\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+y \left (t \right )-3 \end{align*}

✓ Maple. Time used: 0.150 (sec). Leaf size: 44

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

\begin{align*} x \left (t \right ) &= -2+{\mathrm e}^{t} \left (\cos \left (2 t \right ) c_1 +\sin \left (2 t \right ) c_2 \right ) \\ y \left (t \right ) &= -1+{\mathrm e}^{t} \left (\cos \left (2 t \right ) c_2 -\sin \left (2 t \right ) c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.19 (sec). Leaf size: 184

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

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

✓ Sympy. Time used: 0.291 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) - 2*y(t) + Derivative(x(t), t) - 4,0),Eq(2*x(t) - y(t) + Derivative(y(t), t) + 3,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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