46.10.32 problem 35

Internal problem ID [9712]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. EXERCISES 8.2. Page 346
Problem number : 35
Date solved : Tuesday, September 30, 2025 at 06:32:13 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.138 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 51

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

\begin{align*} x(t)&\to e^{4 t} (c_1 \cos (t)+(c_1+c_2) \sin (t))\\ y(t)&\to e^{4 t} (c_2 \cos (t)-(2 c_1+c_2) \sin (t)) \end{align*}

✓ Sympy. Time used: 0.066 (sec). Leaf size: 54

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

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