82.55.2 problem Ex. 2

Internal problem ID [18966]
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. 2
Date solved : Monday, March 31, 2025 at 06:26:49 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-7 x \left (t \right )+y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )-2 x \left (t \right )-5 y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 44

ode:=[diff(x(t),t)-7*x(t)+y(t) = 0, diff(y(t),t)-2*x(t)-5*y(t) = 0]; 
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 ) &= {\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 ) \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 53

ode={D[x[t],t]-7*x[t]+y[t]==0,D[y[t],t]-2*x[t]-5*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.116 (sec). Leaf size: 54

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(-2*x(t) - 5*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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