82.55.1 problem Ex. 1

Internal problem ID [18965]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter XI. Ordinary differential equations with more than two variables. problems at page 129
Problem number : Ex. 1
Date solved : Monday, March 31, 2025 at 06:26:48 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )+\frac {d}{d t}y \left (t \right )+2 x \left (t \right )+y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )+5 x \left (t \right )+3 y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 37

ode:=[diff(x(t),t)+diff(y(t),t)+2*x(t)+y(t) = 0, diff(y(t),t)+5*x(t)+3*y(t) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 42

ode={D[x[t],t]+D[y[t],t]+2*x[t]+y[t]==0,D[y[t],t]+5*x[t]+3*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to 2 c_2 \sin (t)+c_1 (3 \sin (t)+\cos (t)) \\ y(t)\to c_2 (\cos (t)-3 \sin (t))-5 c_1 \sin (t) \\ \end{align*}

✓ Sympy. Time used: 0.092 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(2*x(t) + y(t) + Derivative(x(t), t) + Derivative(y(t), t),0),Eq(5*x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \left (\frac {C_{1}}{5} + \frac {3 C_{2}}{5}\right ) \sin {\left (t \right )} - \left (\frac {3 C_{1}}{5} - \frac {C_{2}}{5}\right ) \cos {\left (t \right )}, \ y{\left (t \right )} = C_{1} \cos {\left (t \right )} - C_{2} \sin {\left (t \right )}\right ] \]