7.22.8 problem 18

Internal problem ID [583]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.1 (First order systems and applications). Problems at page 335
Problem number : 18
Date solved : Saturday, March 29, 2025 at 04:57:15 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=10 x \left (t \right )-7 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 2\\ y \left (0\right ) = -7 \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 33

ode:=[diff(x(t),t) = -y(t), diff(y(t),t) = 10*x(t)-7*y(t)]; 
ic:=x(0) = 2y(0) = -7; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= \frac {17 \,{\mathrm e}^{-2 t}}{3}-\frac {11 \,{\mathrm e}^{-5 t}}{3} \\ y \left (t \right ) &= \frac {34 \,{\mathrm e}^{-2 t}}{3}-\frac {55 \,{\mathrm e}^{-5 t}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 44

ode={D[x[t],t]==-y[t],D[y[t],t]==10*x[t]-7*y[t]}; 
ic={x[0]==2,y[0]==-7}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-5 t} \left (17 e^{3 t}-11\right ) \\ y(t)\to \frac {1}{3} e^{-5 t} \left (34 e^{3 t}-55\right ) \\ \end{align*}

✓ Sympy. Time used: 0.135 (sec). Leaf size: 34

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(y(t) + Derivative(x(t), t),0),Eq(-10*x(t) + 7*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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