85.88.4 problem 1 (d)

Internal problem ID [23031]
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 499
Problem number : 1 (d)
Date solved : Thursday, October 02, 2025 at 09:17:38 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 )&=0\\ 3 x \left (t \right )+2 \frac {d}{d t}y \left (t \right )+y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 34

ode:=[diff(x(t),t)-2*x(t)+4*y(t) = 0, 3*x(t)+2*diff(y(t),t)+y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t}+c_2 \,{\mathrm e}^{\frac {7 t}{2}} \\ y \left (t \right ) &= c_1 \,{\mathrm e}^{-2 t}-\frac {3 c_2 \,{\mathrm e}^{\frac {7 t}{2}}}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 82

ode={D[x[t],{t,1}]-2*x[t]+4*y[t]==0, 3*x[t]+ 2*D[y[t],{t,1}]+y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{11} e^{-2 t} \left (c_1 \left (8 e^{11 t/2}+3\right )-8 c_2 \left (e^{11 t/2}-1\right )\right )\\ y(t)&\to \frac {1}{11} e^{-2 t} \left (c_2 \left (3 e^{11 t/2}+8\right )-3 c_1 \left (e^{11 t/2}-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.070 (sec). Leaf size: 37

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(3*x(t) + y(t) + 2*Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{- 2 t} - \frac {8 C_{2} e^{\frac {7 t}{2}}}{3}, \ y{\left (t \right )} = C_{1} e^{- 2 t} + C_{2} e^{\frac {7 t}{2}}\right ] \]