56.1.74 problem 74

Internal problem ID [8786]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 74
Date solved : Wednesday, March 05, 2025 at 06:48:27 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=7 x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-4 x \left (t \right )+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 34

ode:=[diff(x(t),t) = 7*x(t)+y(t), diff(y(t),t) = -4*x(t)+3*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{5 t} \left (c_{2} t +c_{1} \right ) \\ y \left (t \right ) &= -{\mathrm e}^{5 t} \left (2 c_{2} t +2 c_{1} -c_{2} \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 45

ode={D[x[t],t]== 7*x[t]+y[t],D[y[t],t] == -4*x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{5 t} (2 c_1 t+c_2 t+c_1) \\ y(t)\to e^{5 t} (c_2-2 (2 c_1+c_2) t) \\ \end{align*}

✓ Sympy. Time used: 0.100 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-7*x(t) - y(t) + Derivative(x(t), t),0),Eq(4*x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = 2 C_{1} t e^{5 t} + \left (C_{1} + 2 C_{2}\right ) e^{5 t}, \ y{\left (t \right )} = - 4 C_{1} t e^{5 t} - 4 C_{2} e^{5 t}\right ] \]