60.9.9 problem 1864

Internal problem ID [11788]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1864
Date solved : Sunday, March 30, 2025 at 09:15:20 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-5 x \left (t \right )-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-7 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.125 (sec). Leaf size: 45

ode:={diff(x(t),t) = -5*x(t)-2*y(t), diff(y(t),t) = x(t)-7*y(t)}; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-6 t} \left (\sin \left (t \right ) c_1 +\cos \left (t \right ) c_2 \right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-6 t} \left (\sin \left (t \right ) c_1 +\sin \left (t \right ) c_2 -\cos \left (t \right ) c_1 +\cos \left (t \right ) c_2 \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 52

ode={D[x[t],t]==-5*x[t]-2*y[t],D[y[t],t]==x[t]-7*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{-6 t} (c_1 \cos (t)+(c_1-2 c_2) \sin (t)) \\ y(t)\to e^{-6 t} (c_2 \cos (t)+(c_1-c_2) \sin (t)) \\ \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(5*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-x(t) + 7*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (C_{1} - C_{2}\right ) e^{- 6 t} \cos {\left (t \right )} - \left (C_{1} + C_{2}\right ) e^{- 6 t} \sin {\left (t \right )}, \ y{\left (t \right )} = C_{1} e^{- 6 t} \cos {\left (t \right )} - C_{2} e^{- 6 t} \sin {\left (t \right )}\right ] \]