72.11.14 problem 26

Internal problem ID [14768]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.4 page 310
Problem number : 26
Date solved : Thursday, March 13, 2025 at 04:18:36 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-3 x \left (t \right )+10 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-x \left (t \right )+3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.038 (sec). Leaf size: 37

ode:=[diff(x(t),t) = -3*x(t)+10*y(t), diff(y(t),t) = -x(t)+3*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_{1} \sin \left (t \right )+c_{2} \cos \left (t \right ) \\ y &= \frac {\cos \left (t \right ) c_{1}}{10}-\frac {c_{2} \sin \left (t \right )}{10}+\frac {3 c_{1} \sin \left (t \right )}{10}+\frac {3 c_{2} \cos \left (t \right )}{10} \\ \end{align*}

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

ode={D[x[t],t]==-3*x[t]+10*y[t],D[y[t],t]==-x[t]+3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to 10 c_2 \sin (t)+c_1 (\cos (t)-3 \sin (t)) \\ y(t)\to c_2 (3 \sin (t)+\cos (t))-c_1 \sin (t) \\ \end{align*}

✓ Sympy. Time used: 0.080 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) - 10*y(t) + Derivative(x(t), t),0),Eq(x(t) - 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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