71.19.8 problem 8

Internal problem ID [14529]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 10. Applications of Systems of Equations. Exercises 10.2 page 432
Problem number : 8
Date solved : Monday, March 31, 2025 at 12:29:39 PM
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 )-6\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )-y \left (t \right )+2 \end{align*}

✓ Maple. Time used: 0.147 (sec). Leaf size: 56

ode:=[diff(x(t),t) = 3*x(t)-2*y(t)-6, diff(y(t),t) = 4*x(t)-y(t)+2]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -2+{\mathrm e}^{t} \left (\sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 \right ) \\ y \left (t \right ) &= -6+{\mathrm e}^{t} \left (\sin \left (2 t \right ) c_1 +\sin \left (2 t \right ) c_2 +\cos \left (2 t \right ) c_1 -\cos \left (2 t \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.199 (sec). Leaf size: 208

ode={D[x[t],t]==3*x[t]-2*y[t]-6,D[y[t],t]==4*x[t]-1*y[t]+2}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^t \left (-\sin (2 t) \int _1^t2 e^{-K[2]} (\cos (2 K[2])+7 \sin (2 K[2]))dK[2]+(\sin (2 t)+\cos (2 t)) \int _1^te^{-K[1]} (8 \sin (2 K[1])-6 \cos (2 K[1]))dK[1]-c_2 \sin (2 t)+c_1 (\sin (2 t)+\cos (2 t))\right ) \\ y(t)\to e^t \left ((\cos (2 t)-\sin (2 t)) \int _1^t2 e^{-K[2]} (\cos (2 K[2])+7 \sin (2 K[2]))dK[2]+2 \sin (2 t) \int _1^te^{-K[1]} (8 \sin (2 K[1])-6 \cos (2 K[1]))dK[1]+2 c_1 \sin (2 t)+c_2 (\cos (2 t)-\sin (2 t))\right ) \\ \end{align*}

✓ Sympy. Time used: 0.321 (sec). Leaf size: 88

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 2*y(t) + Derivative(x(t), t) + 6,0),Eq(-4*x(t) + y(t) + Derivative(y(t), t) - 2,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {C_{1}}{2} - \frac {C_{2}}{2}\right ) e^{t} \cos {\left (2 t \right )} - \left (\frac {C_{1}}{2} + \frac {C_{2}}{2}\right ) e^{t} \sin {\left (2 t \right )} - 2 \sin ^{2}{\left (2 t \right )} - 2 \cos ^{2}{\left (2 t \right )}, \ y{\left (t \right )} = C_{1} e^{t} \cos {\left (2 t \right )} - C_{2} e^{t} \sin {\left (2 t \right )} - 6 \sin ^{2}{\left (2 t \right )} - 6 \cos ^{2}{\left (2 t \right )}\right ] \]