52.10.28 problem 29

Internal problem ID [8422]
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 : 29
Date solved : Sunday, March 30, 2025 at 01:04:29 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+4 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+6 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = -1\\ y \left (0\right ) = 6 \end{align*}

✓ Maple. Time used: 0.119 (sec). Leaf size: 28

ode:=[diff(x(t),t) = 2*x(t)+4*y(t), diff(y(t),t) = -x(t)+6*y(t)]; 
ic:=x(0) = -1y(0) = 6; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{4 t} \left (26 t -1\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{4 t} \left (52 t +24\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 30

ode={D[x[t],t]==2*x[t]+4*y[t],D[y[t],t]==-x[t]+6*y[t]}; 
ic={x[0]==-1,y[0]==6}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{4 t} (26 t-1) \\ y(t)\to e^{4 t} (13 t+6) \\ \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 42

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

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