66.11.3 problem 5

Internal problem ID [16123]
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 : 5
Date solved : Thursday, October 02, 2025 at 10:42:41 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )-5 y\\ y^{\prime }&=3 x \left (t \right )+y \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=4 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.182 (sec). Leaf size: 47

ode:=[diff(x(t),t) = -3*x(t)-5*y(t), diff(y(t),t) = 3*x(t)+y(t)]; 
ic:=[x(0) = 4, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} \left (-\frac {8 \sqrt {11}\, \sin \left (\sqrt {11}\, t \right )}{11}+4 \cos \left (\sqrt {11}\, t \right )\right ) \\ y \left (t \right ) &= \frac {12 \,{\mathrm e}^{-t} \sqrt {11}\, \sin \left (\sqrt {11}\, t \right )}{11} \\ \end{align*}

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 63

ode={D[x[t],t]==-3*x[t]-5*y[t],D[y[t],t]==3*x[t]+y[t]}; 
ic={x[0]==4,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {4}{11} e^{-t} \left (11 \cos \left (\sqrt {11} t\right )-2 \sqrt {11} \sin \left (\sqrt {11} t\right )\right )\\ y(t)&\to \frac {12 e^{-t} \sin \left (\sqrt {11} t\right )}{\sqrt {11}} \end{align*}

✓ Sympy. Time used: 0.110 (sec). Leaf size: 83

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

\[ \left [ x{\left (t \right )} = - \left (\frac {2 C_{1}}{3} + \frac {\sqrt {11} C_{2}}{3}\right ) e^{- t} \cos {\left (\sqrt {11} t \right )} - \left (\frac {\sqrt {11} C_{1}}{3} - \frac {2 C_{2}}{3}\right ) e^{- t} \sin {\left (\sqrt {11} t \right )}, \ y{\left (t \right )} = C_{1} e^{- t} \cos {\left (\sqrt {11} t \right )} - C_{2} e^{- t} \sin {\left (\sqrt {11} t \right )}\right ] \]