72.11.13 problem 24

Internal problem ID [14767]
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 : 24
Date solved : Thursday, March 13, 2025 at 04:18:35 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-\frac {9 x \left (t \right )}{10}-2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+\frac {11 y \left (t \right )}{10} \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 34

ode:=[diff(x(t),t) = -9/10*x(t)-2*y(t), diff(y(t),t) = x(t)+11/10*y(t)]; 
ic:=x(0) = 1y(0) = 1; 
dsolve([ode,ic]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {t}{10}} \left (-3 \sin \left (t \right )+\cos \left (t \right )\right ) \\ y &= -\frac {{\mathrm e}^{\frac {t}{10}} \left (-4 \sin \left (t \right )-2 \cos \left (t \right )\right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 38

ode={D[x[t],t]==-9/10*x[t]-2*y[t],D[y[t],t]==x[t]+11/10*y[t]}; 
ic={x[0]==1,y[0]==1}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to e^{t/10} (\cos (t)-3 \sin (t)) \\ y(t)\to e^{t/10} (2 \sin (t)+\cos (t)) \\ \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 49

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(9*x(t)/10 + 2*y(t) + Derivative(x(t), t),0),Eq(-x(t) - 11*y(t)/10 + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \left (C_{1} - C_{2}\right ) e^{\frac {t}{10}} \sin {\left (t \right )} - \left (C_{1} + C_{2}\right ) e^{\frac {t}{10}} \cos {\left (t \right )}, \ y{\left (t \right )} = C_{1} e^{\frac {t}{10}} \cos {\left (t \right )} - C_{2} e^{\frac {t}{10}} \sin {\left (t \right )}\right ] \]