78.5.7 problem 5.a

Internal problem ID [21046]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 6, Linear systems. Problems section 6.9
Problem number : 5.a
Date solved : Thursday, October 02, 2025 at 07:01:44 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=3 x \left (t \right )-y\\ y^{\prime }&=2 x \left (t \right )-2 y \end{align*}

✓ Maple. Time used: 0.103 (sec). Leaf size: 85

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (1+\sqrt {17}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {17}\right ) t}{2}} \\ y \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{\frac {\left (1+\sqrt {17}\right ) t}{2}} \sqrt {17}}{2}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {17}\right ) t}{2}} \sqrt {17}}{2}+\frac {5 c_1 \,{\mathrm e}^{\frac {\left (1+\sqrt {17}\right ) t}{2}}}{2}+\frac {5 c_2 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {17}\right ) t}{2}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 153

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

\begin{align*} x(t)&\to \frac {1}{34} e^{\frac {1}{2} \left (t-\sqrt {17} t\right )} \left (c_1 \left (\left (17+5 \sqrt {17}\right ) e^{\sqrt {17} t}+17-5 \sqrt {17}\right )-2 \sqrt {17} c_2 \left (e^{\sqrt {17} t}-1\right )\right )\\ y(t)&\to \frac {1}{34} e^{\frac {1}{2} \left (t-\sqrt {17} t\right )} \left (4 \sqrt {17} c_1 \left (e^{\sqrt {17} t}-1\right )-c_2 \left (\left (5 \sqrt {17}-17\right ) e^{\sqrt {17} t}-17-5 \sqrt {17}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 75

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

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (5 - \sqrt {17}\right ) e^{\frac {t \left (1 - \sqrt {17}\right )}{2}}}{4} + \frac {C_{2} \left (\sqrt {17} + 5\right ) e^{\frac {t \left (1 + \sqrt {17}\right )}{2}}}{4}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (1 - \sqrt {17}\right )}{2}} + C_{2} e^{\frac {t \left (1 + \sqrt {17}\right )}{2}}\right ] \]