204 Ordinary Differential Equations
W(y1, yz, ... , Yn, x) =
A judicious choice for x in the Wronskian above is x = 0. Making this choice,
we find
1 1 1
r1 r2 rn
W(y1,y2, ... ,yn,O) =
n-1
r1
r2 n-1 ... n-1
rnsince the roots are distinct. The last determinant in the calculation above is
known as the Vandermonde determinant and its value is well known.
For example, to find the general solution of the third order homogeneous
linear differential equation
(7) 2y"' - y" - 2y' + y = 0,
we write the auxiliary equation
p(r) = 2r^3 -r^2 -2r+1=0.
Factoring, we find
p(r) = (r + l)(r - 1)(2r - 1) = 0.
So the roots of the a uxiliary equation are -1, 1, and ~, and three linearly
independent solutions of the DE (7) are
Y1(x) = e-x, Y2(x) =ex, and y3(x) = ex/^2.Consequently, the general solution of the DE (7) is
y(x) = C1e-x + c2ex + c3exl^2
where c1, c2, and c3 are arbitrary constants.
Repeated Real Roots Consider the differential equation(8) y^111 - 6y^11 + 12y' - 8y = 0.
The auxiliary equation is