66.10.8 problem 8

Internal problem ID [16106]
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 3.2. page 277
Problem number : 8
Date solved : Thursday, October 02, 2025 at 10:42:31 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.107 (sec). Leaf size: 85

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (3+\sqrt {5}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}} \\ y \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{\frac {\left (3+\sqrt {5}\right ) t}{2}} \sqrt {5}}{2}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}} \sqrt {5}}{2}+\frac {c_1 \,{\mathrm e}^{\frac {\left (3+\sqrt {5}\right ) t}{2}}}{2}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (\sqrt {5}-3\right ) t}{2}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 144

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

\begin{align*} x(t)&\to \frac {1}{10} e^{-\frac {1}{2} \left (\sqrt {5}-3\right ) t} \left (c_1 \left (\left (5+\sqrt {5}\right ) e^{\sqrt {5} t}+5-\sqrt {5}\right )-2 \sqrt {5} c_2 \left (e^{\sqrt {5} t}-1\right )\right )\\ y(t)&\to -\frac {1}{10} e^{-\frac {1}{2} \left (\sqrt {5}-3\right ) t} \left (2 \sqrt {5} c_1 \left (e^{\sqrt {5} t}-1\right )+c_2 \left (\left (\sqrt {5}-5\right ) e^{\sqrt {5} t}-5-\sqrt {5}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 76

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

\[ \left [ x{\left (t \right )} = - \frac {C_{1} \left (1 - \sqrt {5}\right ) e^{\frac {t \left (3 - \sqrt {5}\right )}{2}}}{2} - \frac {C_{2} \left (1 + \sqrt {5}\right ) e^{\frac {t \left (\sqrt {5} + 3\right )}{2}}}{2}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (3 - \sqrt {5}\right )}{2}} + C_{2} e^{\frac {t \left (\sqrt {5} + 3\right )}{2}}\right ] \]