Ordinary differential equation
Solve the ode y'' - y' +4y = 0 as a system of first order odes.
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SOLUTION This solution is FREE courtesy of BrainMass!
set x_1 = y and x_2 = y' so that
x_1 ' = y' and
x_2 ' = y '' = y ' - 4y = x_2 - 4x_1
hence the given equation is equivalent to the system
X ' =
(0 1)
(-4 1) X
where X = [x_1 x_2] is a column vector.
The eigenvalues of the matrix A on the right side are
(1/2)(1 +/- i.sqrt(15))
and an eigenvector corresponding to the " + " eigenvalue is
[ 1 - i.sqrt(15), 8]
so the solution we get from this eigenvalue/eigenvector pair is
X(t) = e^( (1/2 + i.(sqrt(15)/2)) t } [1 - i.sqrt(15) 8]
= e^{(1/2)t} (cos ((sqrt(15)/2) t) + i. sin ((sqrt(15)/2) t)) [ 1 - i.sqrt(15) 8]
which can be written in the form U(t) + i. V(t) where
U(t) = [e^{(1/2) t} (cos (sqrt(15)/2) t) + sqrt(15). sin (sqrt(15)/2) t); 8e^{(1/2) t} cos( sqrt(15)/2) t)]
V(t) = [e^{(1/2)t} (sin (sqrt(15)/2) t) -- sqrt(15). cos (sqrt(15)/2) t); 8e^{(1/2)t} sin (sqrt(15)/2) t)]
Finally, since U and V form a linearly independent solution set, the general solution to the system X ' = AX is
X(t) = c_1 U(t) + c_2 V(t).
© BrainMass Inc. brainmass.com December 24, 2021, 7:44 pm ad1c9bdddf>https://brainmass.com/math/ordinary-differential-equations/ordinary-differential-equation-example-problem-215011