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In the earlier section we introduced the Wronskian to assist us find out whether two solutions were a fundamental set of solutions. Under this section we will look at the other application of the Wronskian and also an alternate method of computing the Wronskian.
Let's begin with the application. We require introducing a couple of new concepts first.
Specified two non-zero functions f(x) and g(x) write down the subsequent equation
c f ( x ) + k g ( x ) = 0
See that c = 0 and k = 0 will make (1) true for all x regardless of the functions which we use.
Here, if we can get non-zero constants c and k for that (1) will also be true for all x so we call the two functions linearly dependent. Conversely, if the only two constants for that (1) is true are c = 0 and k = 0 so we call the functions linearly independent.
Let G be a group acting on a set X. The action is called faithful if for any g ≠ 1 ∈ G there exists an x ∈ X such that gx ≠ x. That is, only the identity fixes everything. Prov
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