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The Wronskian

The Wronskian

Given an inhomogeneous linear second-order ode:

¨x+p(t)˙x+q(t)x=0

Suppose 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)=u0

We 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)|0

The Wronskian W is given by:

W=|X1(t0)X2(t0)˙X1(t0)˙X2(t0)|=X1(t0)˙X2(t0)˙X1(t0)X2(t0)0

Example

˙X1(t)=αeαt˙X2(t)=βeβtW=|eαteβtαeαtβeβt|=βeαteβtαeαteβt

Thus, W0 when αβ.