44.29.7 problem 3(c)

Internal problem ID [9489]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page 400
Problem number : 3(c)
Date solved : Tuesday, September 30, 2025 at 06:19:23 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.105 (sec). Leaf size: 85

ode:=[diff(x(t),t) = 3*x(t)-5*y(t), diff(y(t),t) = -x(t)+2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}} \\ y \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}} \sqrt {21}}{10}+\frac {c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {21}\right ) t}{2}}}{10}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {21}\right ) t}{2}}}{10} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 144

ode={D[x[t],t]==3*x[t]-5*y[t],D[y[t],t]==-x[t]+2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{42} e^{-\frac {1}{2} \left (\sqrt {21}-5\right ) t} \left (c_1 \left (\left (21+\sqrt {21}\right ) e^{\sqrt {21} t}+21-\sqrt {21}\right )-10 \sqrt {21} c_2 \left (e^{\sqrt {21} t}-1\right )\right )\\ y(t)&\to -\frac {1}{42} e^{-\frac {1}{2} \left (\sqrt {21}-5\right ) t} \left (2 \sqrt {21} c_1 \left (e^{\sqrt {21} t}-1\right )+c_2 \left (\left (\sqrt {21}-21\right ) e^{\sqrt {21} t}-21-\sqrt {21}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 76

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {21}\right ) e^{\frac {t \left (5 - \sqrt {21}\right )}{2}}}{2} - \frac {C_{2} \left (1 + \sqrt {21}\right ) e^{\frac {t \left (\sqrt {21} + 5\right )}{2}}}{2}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (5 - \sqrt {21}\right )}{2}} + C_{2} e^{\frac {t \left (\sqrt {21} + 5\right )}{2}}\right ] \]