66.11.4 problem 6

Internal problem ID [16124]
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 : 6
Date solved : Thursday, October 02, 2025 at 10:42:42 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=2 y\\ y^{\prime }&=-2 x \left (t \right )-y \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.160 (sec). Leaf size: 62

ode:=[diff(x(t),t) = 2*y(t), diff(y(t),t) = -2*x(t)-y(t)]; 
ic:=[x(0) = -1, y(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (\frac {\sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{2}\right )}{5}-\cos \left (\frac {\sqrt {15}\, t}{2}\right )\right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {t}{2}} \left (-\frac {4 \sqrt {15}\, \sin \left (\frac {\sqrt {15}\, t}{2}\right )}{5}-4 \cos \left (\frac {\sqrt {15}\, t}{2}\right )\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 92

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

\begin{align*} x(t)&\to \frac {1}{5} e^{-t/2} \left (\sqrt {15} \sin \left (\frac {\sqrt {15} t}{2}\right )-5 \cos \left (\frac {\sqrt {15} t}{2}\right )\right )\\ y(t)&\to \frac {1}{5} e^{-t/2} \left (\sqrt {15} \sin \left (\frac {\sqrt {15} t}{2}\right )+5 \cos \left (\frac {\sqrt {15} t}{2}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.112 (sec). Leaf size: 92

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

\[ \left [ x{\left (t \right )} = - \left (\frac {C_{1}}{4} - \frac {\sqrt {15} C_{2}}{4}\right ) e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {15} t}{2} \right )} + \left (\frac {\sqrt {15} C_{1}}{4} + \frac {C_{2}}{4}\right ) e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {15} t}{2} \right )}, \ y{\left (t \right )} = C_{1} e^{- \frac {t}{2}} \cos {\left (\frac {\sqrt {15} t}{2} \right )} - C_{2} e^{- \frac {t}{2}} \sin {\left (\frac {\sqrt {15} t}{2} \right )}\right ] \]