85.83.6 problem 1 (f)

Internal problem ID [23003]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 444
Problem number : 1 (f)
Date solved : Thursday, October 02, 2025 at 09:17:26 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y \left (t \right )+6 y \left (t \right )&=\frac {d}{d t}x \left (t \right )\\ 3 x \left (t \right )-\frac {d}{d t}x \left (t \right )&=2 \frac {d}{d t}y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=2 \\ y \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 31

ode:=[diff(y(t),t)+6*y(t) = diff(x(t),t), 3*x(t)-diff(x(t),t) = 2*diff(y(t),t)]; 
ic:=[x(0) = 2, y(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= 4 \,{\mathrm e}^{2 t}-2 \,{\mathrm e}^{-3 t} \\ y \left (t \right ) &= {\mathrm e}^{2 t}+2 \,{\mathrm e}^{-3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 36

ode={D[y[t],{t,1}]+6*y[t]==D[x[t],t],3*x[t]-D[x[t],{t,1}]==2*D[y[t],t]}; 
ic={x[0]==2,y[0]==3}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^{-3 t} \left (4 e^{5 t}-2\right )\\ y(t)&\to e^{-3 t} \left (e^{5 t}+2\right ) \end{align*}

✓ Sympy. Time used: 0.086 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(6*y(t) - Derivative(x(t), t) + Derivative(y(t), t),0),Eq(3*x(t) - Derivative(x(t), t) - 2*Derivative(y(t), t),0)] 
ics = {x(0): 2, y(0): 3} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 4 e^{2 t} - 2 e^{- 3 t}, \ y{\left (t \right )} = e^{2 t} + 2 e^{- 3 t}\right ] \]