Ma316-001-F10 further explanation, §3.2

further explanation

existence and uniqueness

Consider two functions p(t) and q(t):

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We know that the differential equation
y '' + p(t)  y ' + q(t) y = 0,
with initial conditions
,

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Has a unique solution:

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...as long as p(t) and q(t) are continuous

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superposition of two (linearly independent) solutions can satisfy any initial condition

Now, suppose that we have two solutions,   and , of
y '' + p(t) y ' + q(t) y = 0

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Then, if their Wronskian is non-zero at , given an initial condition there:

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we can find a linear combination of and ,   that solves the initial value problem
y '' + p(t) y ' + q(t) y = 0,  ,  :

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superposition of two (linearly independent) solutions gives the general solution

Now suppose that we are given any  solution φ(t) of the differential equation
y '' + p(t) y ' + q(t) y = 0.
We want to show that we can write φ(t) as a linear combination of and , that is, that we can find a and so that

So, suppose that we have such a φ  and that we know that at some point the Wronskian of and is non-zero.

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Then, at , we know that φ(t)  satisfies the initial value problem

That is, a solution to the differential equation with the initial conditions:

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That is, we're solving the differential equation with these initial conditions:

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But, by the work in the previous section, we know that we can find a and a such that

satisfies the initial value problem
.
Thus:

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And the uniqueness theorem (the first section above) says that this solution is unique—that is, it must be the same function as the φ(t) that we started with:

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Thus, because we started off with the assumption that φ(t)  was any  solution to the differential equation, we've shown that the linear combination

is the general solution  to the differential equation!

Super cool!