60.9.7 problem 1862

Internal problem ID [11786]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1862
Date solved : Sunday, March 30, 2025 at 09:15:17 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.124 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 46

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

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

✓ Sympy. Time used: 0.106 (sec). Leaf size: 49

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

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