57.20.3 problem 2(c)

Internal problem ID [14506]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 218
Problem number : 2(c)
Date solved : Thursday, October 02, 2025 at 09:37:58 AM
CAS classification : system_of_ODEs

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

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

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

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

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 145

ode={D[x[t],t]==-x[t]+y[t],D[y[t],t]==x[t]-2*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 (3+\sqrt {5}\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 (3+\sqrt {5}\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: 0.116 (sec). Leaf size: 75

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

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