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The Wronskian
The Wronskian
Given an inhomogeneous linear second-order ode:
¨x+p(t)˙x+q(t)x=0Suppose that x=X1(t) and x=X2(t) are two solutions to the ode.
According to the principle of superposition, we can write the general solution to the ode as:
x=c1X1(t)+c2X2(t)To fulfill a given initial condition
x(t0)=x0,˙x(t0)=u0We need to solve a system of linear equations:
c1X1(t0)+c2X2(t0)=x0(1)c1˙X1(t0)+c2˙X2(t0)=u0(2)To solve (1) and (2), we need to find the RREF of:
[X1(t0)X2(t0)x0˙X1(t0)˙X2(t0)u0]Thus, we need to find the inverse of:
[X1(t0)X2(t0)˙X1(t0)˙X2(t0)]The inverse of the matrix exist when:
|X1(t0)X2(t0)˙X1(t0)˙X2(t0)|≠0The Wronskian W is given by:
W=|X1(t0)X2(t0)˙X1(t0)˙X2(t0)|=X1(t0)˙X2(t0)−˙X1(t0)X2(t0)≠0Example
Thus, W≠0 when α≠β.