56.1.73 problem 73

Internal problem ID [8785]
Book : Own collection of miscellaneous problems
Section : section 1.0
Problem number : 73
Date solved : Sunday, March 30, 2025 at 01:35:46 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=2 x \left (t \right )+5 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.110 (sec). Leaf size: 32

ode:=[diff(x(t),t) = x(t)-2*y(t), diff(y(t),t) = 2*x(t)+5*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 46

ode={D[x[t],t]== x[t]-2*y[t],D[y[t],t] == 2*x[t]+5*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to -e^{3 t} (c_1 (2 t-1)+2 c_2 t) \\ y(t)\to e^{3 t} (2 (c_1+c_2) t+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.103 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) - 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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