76.8.15 problem 15

Internal problem ID [17433]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.4 (Complex Eigenvalues). Problems at page 177
Problem number : 15
Date solved : Monday, March 31, 2025 at 04:13:39 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-5 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=a x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.148 (sec). Leaf size: 99

ode:=[diff(x(t),t) = 2*x(t)-5*y(t), diff(y(t),t) = a*x(t)-2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (\sqrt {5 a -4}\, t \right )+c_2 \cos \left (\sqrt {5 a -4}\, t \right ) \\ y \left (t \right ) &= -\frac {c_1 \sqrt {5 a -4}\, \cos \left (\sqrt {5 a -4}\, t \right )}{5}+\frac {c_2 \sqrt {5 a -4}\, \sin \left (\sqrt {5 a -4}\, t \right )}{5}+\frac {2 c_1 \sin \left (\sqrt {5 a -4}\, t \right )}{5}+\frac {2 c_2 \cos \left (\sqrt {5 a -4}\, t \right )}{5} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 184

ode={D[x[t],t]==2*x[t]-5*y[t],D[y[t],t]==a*x[t]-2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {e^{-\sqrt {4-5 a} t} \left (c_1 \left (\left (\sqrt {4-5 a}+2\right ) e^{2 \sqrt {4-5 a} t}+\sqrt {4-5 a}-2\right )-5 c_2 \left (e^{2 \sqrt {4-5 a} t}-1\right )\right )}{2 \sqrt {4-5 a}} \\ y(t)\to \frac {e^{-\sqrt {4-5 a} t} \left (a c_1 \left (e^{2 \sqrt {4-5 a} t}-1\right )+c_2 \left (\left (\sqrt {4-5 a}-2\right ) e^{2 \sqrt {4-5 a} t}+\sqrt {4-5 a}+2\right )\right )}{2 \sqrt {4-5 a}} \\ \end{align*}

✓ Sympy. Time used: 0.219 (sec). Leaf size: 82

from sympy import * 
t = symbols("t") 
a = symbols("a") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-2*x(t) + 5*y(t) + Derivative(x(t), t),0),Eq(-a*x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {5 C_{1} e^{t \sqrt {4 - 5 a}}}{\sqrt {4 - 5 a} - 2} + \frac {5 C_{2} e^{- t \sqrt {4 - 5 a}}}{\sqrt {4 - 5 a} + 2}, \ y{\left (t \right )} = C_{1} e^{t \sqrt {4 - 5 a}} + C_{2} e^{- t \sqrt {4 - 5 a}}\right ] \]