80.8.3 problem 3

Internal problem ID [21360]
Book : A Textbook on Ordinary Differential Equations by Shair Ahmad and Antonio Ambrosetti. Second edition. ISBN 978-3-319-16407-6. Springer 2015
Section : Chapter 8. Qualitative analysis of 2 by 2 systems and nonlinear second order equations. Excercise 8.5 at page 184
Problem number : 3
Date solved : Thursday, October 02, 2025 at 07:28:45 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 217

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

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {t \left (\sqrt {-8+\left (a -b \right )^{2}}+a +b \right )}{2}}+c_2 \,{\mathrm e}^{-\frac {t \left (\sqrt {-8+\left (a -b \right )^{2}}-a -b \right )}{2}} \\ y \left (t \right ) &= -\frac {c_2 \,{\mathrm e}^{-\frac {t \left (\sqrt {-8+\left (a -b \right )^{2}}-a -b \right )}{2}} a}{2}-\frac {c_1 \,{\mathrm e}^{\frac {t \left (\sqrt {-8+\left (a -b \right )^{2}}+a +b \right )}{2}} a}{2}+\frac {c_1 \,{\mathrm e}^{\frac {t \left (\sqrt {-8+\left (a -b \right )^{2}}+a +b \right )}{2}} \sqrt {-8+\left (a -b \right )^{2}}}{2}+\frac {c_1 \,{\mathrm e}^{\frac {t \left (\sqrt {-8+\left (a -b \right )^{2}}+a +b \right )}{2}} b}{2}-\frac {c_2 \,{\mathrm e}^{-\frac {t \left (\sqrt {-8+\left (a -b \right )^{2}}-a -b \right )}{2}} \sqrt {-8+\left (a -b \right )^{2}}}{2}+\frac {c_2 \,{\mathrm e}^{-\frac {t \left (\sqrt {-8+\left (a -b \right )^{2}}-a -b \right )}{2}} b}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 362

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

\begin{align*} x(t)&\to \frac {e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2-8}+a+b\right )} \left (c_1 \sqrt {a^2-2 a b+b^2-8} e^{t \sqrt {a^2-2 a b+b^2-8}}+a c_1 \left (e^{t \sqrt {a^2-2 a b+b^2-8}}-1\right )-b c_1 \left (e^{t \sqrt {a^2-2 a b+b^2-8}}-1\right )+2 c_2 e^{t \sqrt {a^2-2 a b+b^2-8}}+c_1 \sqrt {a^2-2 a b+b^2-8}-2 c_2\right )}{2 \sqrt {a^2-2 a b+b^2-8}}\\ y(t)&\to \frac {e^{\frac {1}{2} t \left (-\sqrt {a^2-2 a b+b^2-8}+a+b\right )} \left (c_2 \left (a \left (-e^{t \sqrt {a^2-2 a b+b^2-8}}\right )+b \left (e^{t \sqrt {a^2-2 a b+b^2-8}}-1\right )+\sqrt {a^2-2 a b+b^2-8} \left (e^{t \sqrt {a^2-2 a b+b^2-8}}+1\right )+a\right )-4 c_1 \left (e^{t \sqrt {a^2-2 a b+b^2-8}}-1\right )\right )}{2 \sqrt {a^2-2 a b+b^2-8}} \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 156

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

\[ \left [ x{\left (t \right )} = - \frac {C_{1} \left (a - b + \sqrt {a^{2} - 2 a b + b^{2} - 8}\right ) e^{\frac {t \left (a + b + \sqrt {a^{2} - 2 a b + b^{2} - 8}\right )}{2}}}{4} + \frac {C_{2} \left (- a + b + \sqrt {a^{2} - 2 a b + b^{2} - 8}\right ) e^{\frac {t \left (a + b - \sqrt {a^{2} - 2 a b + b^{2} - 8}\right )}{2}}}{4}, \ y{\left (t \right )} = C_{1} e^{\frac {t \left (a + b + \sqrt {a^{2} - 2 a b + b^{2} - 8}\right )}{2}} + C_{2} e^{\frac {t \left (a + b - \sqrt {a^{2} - 2 a b + b^{2} - 8}\right )}{2}}\right ] \]