Download Abel`s theorem and the Wronskian

Review of Lecture 10 –Existence and Uniqueness
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Second order linear equations have the form
P(t)y ′′ + Q(t)y ′ + R(t)y = G (t)
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An initial value problem specifies y0 and y0′
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There is a unique solution to an initial value problem
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It is defined on an interval that extends as long as P(t) 6= 0
and P, Q, R, and G have continuous derivatives.
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The equation is homogeneous if the right side is zero.
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When the coefficients are constant we have the homogeneous
equation
ay ′′ + by ′ + c = 0
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It has solutions e λt if λ is a root of the characteristic equation
aλ2 + bλ + c = 0
Second order linear homogeneous constant-coefficient
equations
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A fundamental set of solutions is a set of two solutions u and
v such that every solution is a linear combination of u and v
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If we believe uniqueness, then u and v form a fundamental set
if we can solve any initial value problem using a linear
combination of u and v .
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In case the characteristic equation has two distinct real
solutions, we can do that, so we have a fundamental set of
solutions.
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That leaves the cases of equal roots and complex roots still to
consider.
The Wronskian
Consider a linear homogeneous equation
y ′′ + p(t)y ′ + q(t)y = 0.
Given two solutions u and v , the Wronskian W (u, v ) is another
function defined by
uv ′ − vu ′
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It is (minus) the numerator of the derivative of u/v , so if u is
a constant multiple of v , then the Wronskian W (u, v ) is
identically zero.
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If u and v agree at some point t0 , and their derivatives also
agree at t0 , then the Wronskian W (u, v ) vanishes at t0 .
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That is, if u and v are two solutions of the same initial value
problem, their Wronskian vanishes at t0 .
The Wronskian satisfies a first-order linear equation
Namely
W ′ + p(t)W = 0.
That is verified just by plugging in the definition of W :
W ′ + p(t)W
= (uv ′ − vu ′ )′ + p(t)(uv ′ − vu ′ )
= u ′ v ′ + uv ′′ − v ′ u ′ − vu ′′ + p(t)uv ′ − p(t)vu ′
= uv ′′ − vu ′′ + p(t)uv ′ − p(t)vu ′
= u(v ′′ + p(t)v ′ ) − v (u ′′ + p(t)u ′ )
= u(−q(t)v ) − v (−q(t)u) since u and v are solutions
= 0
Abel’s theorem
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The uniqueness theorem that we didn’t prove (yet) can be
phrased this way: if u and v solve the same initial value
problem, then they are equal.
Abel’s theorem
◮
The uniqueness theorem that we didn’t prove (yet) can be
phrased this way: if u and v solve the same initial value
problem, then they are equal.
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Abel’s theorem: if W (u, v ) vanishes at one point t0 , then it is
identically zero.
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Abel’s theorem implies uniqueness: Suppose u and v solve the
same initial value problem. Then W (u, v ) vanishes at t0 .
Then by Abel, W (u, v ) is identically zero. Then u is a
constant multiple of v . But u(t0 ) = v (t0 ). Hence u = v
everywhere.
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That is a good reason why Abel’s theorem and the Wronskian
are important.
Proof of Abel’s theorem
We have
W ′ + p(t)W = 0.
We know how to solve linear equations:
dW
= p(t)dt
W
Z
ln W = p(t) dt
W (t) = ce −
R
p(t) dt
But e x is never zero, so the only way W (t) can vanish anywhere is
if it is identically zero.
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This proves Abel’s theorem, and hence also the uniqueness
theorem. I think that is the main point of Abel’s theorem–it
implies uniqueness.
Abel’s theorem implies uniqueness
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The textbook does not state this important point, but instead
says that the existence and uniqueness theorems are too
difficult to prove in this course.
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But at least the uniqueness proof is actually given in the
book, all but the final statement of the conclusion.
The Wronskian and the general solution
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Given a second-order linear equation we want the “general
solution”.
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That means, we want two solutions u and v such that every
solution is a linear combination Au + Bv .
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it suffices if we can solve any initial value problem by choosing
A and B correctly.
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It turns out, we can do that if (and only if) the Wronskian
W (u, v ) is not zero.
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“not zero” could mean “not zero anywhere” or “not zero
everywhere”, since we proved those are equivalent.
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This gives us a good test for whether u and v form a
fundamental set:
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u and v form a fundamental set if and only if W (u, v ) 6= 0.
The algebra to prove that criterion
Suppose given two solutions u and v ; we want to look for a
solution to the initial value problem with y (t0 ) = y0 and
y ′ (t0 ) = y0′ . We take y = Au + Bv . Then let u0 = u(t0 ), etc.
Then
Au0 + Bv0 = y0
Au0′ + Bv0′ = y0′
Those are two linear equations for A and B. They have a solution if
and only if the determinant of the coefficients is not zero. That is
u0 v0′ − v0 u0′ 6= 0.
But the left side is just W (u, v )(t0 ). We have proved:
◮ Every initial value problem can be solved in the form Au + Bv
if and only if W (u, v ) 6= 0.
◮ or in other words, u and v form a fundamental set of solutions
if and only if W (u, v ) 6= 0.
An example: Bessel’s equation
Bessel’s equation is
x 2 y ′′ + xy ′ + (x 2 − n2 )y = 0
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n is a parameter; it’s a different equation for each integer n.
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Since the coefficient of y ′ ′ is zero when x = 0, we haven’t
proved there is a solution defined for x = 0.
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There are solutions passing through, say, (1, y1 ) for a given y1 ,
and they are defined for x > 0, but what happens to them as
x goes to zero?
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They might go to ∞, or to 1, or 0, or maybe they might
oscillate rapidly?
Bessel functions
The Bessel functions (of the first kind) Jn are defined as the
solutions of Bessel’s equation that are not singular (i.e. are
defined) at x = 0. (We haven’t proved there are such solutions,
but we will do so later in the course.)
Bessel functions
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The Bessel functions come up repeatedly in physics and
engineering.
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For example, in describing the modes of vibration of a
“drum-head”, i.e. a circular plate with fixed boundary values
subject to small vibrations.
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There are whole books full of formulas about the Bessel
functions.
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Look them up in Wikipedia or Wolfram MathWorld for a
sample.
Bessel Functions of the Second Kind
These are solutions that are singular at x = 0.
Bessel functions
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For each n, the two solutions Jn and Yn form a fundamental
set of solutions of Bessel’s equation.
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We have proved that a fundamental set exists.
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Assuming that Yn goes to −∞ and Jn stays bounded as x
goes to zero, then their ratio is not constant.
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Hence the Wronskian W (Jn , Yn ) is not zero.
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Hence they form a fundamental set.
Bessel functions
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For each n, the two solutions Jn and Yn form a fundamental
set of solutions of Bessel’s equation.
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We have proved that a fundamental set exists.
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Assuming that Yn goes to −∞ and Jn stays bounded as x
goes to zero, then their ratio is not constant.
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Hence the Wronskian W (Jn , Yn ) is not zero.
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Hence they form a fundamental set.
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We haven’t yet studied the behavior of solutions at a singular
point like x = 0 in this example, but the example illustrates
that it is important to do so.
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We will devote a whole lecture to Bessel’s equation about ten
days before Thanksgiving, using infinite series.
Airy functions
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The Airy functions Ai and Bi are defined as two fundamental
solutions of
y ′′ − xy = 0
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Ai (x) tends to 0 for large x.
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Ai is the name of the function, it is not a typo for Ai .
Existence of solutions of the Airy equation?
y ′′ − xy = 0
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The coefficient of y ′′ is 1.
The coefficient of y is x, which is everywhere smooth.
Can we prove the existence of a fundamental solution set?
Existence of solutions of the Airy equation
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The coefficient of y ′′ is 1.
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The coefficient of y is x, which is everywhere smooth.
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Hence every initial value problem has a solution.
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Fix two solutions u and v with different slopes at x = 0 but
u(0) = v (0).
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Then W (u, v ) 6= 0, so u and v form a fundamental set.
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But that doesn’t tell us about the behavior at ∞.
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So we still haven’t got the Airy functions.
Existence of Airy functions
In 1838 Airy defined a diffraction integral representing for each
color the variation of the electromagnetic wave across a rainbow.
Z
1 ∞
1 3
Ai (x) =
t + xt dt
cos
π 0
3
According to Wikipedia, differentiation under the integral sign
shows that Ai satisfies the Airy equation
y ′′ − xy = 0.
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That exercise is not very straightforward, as one has a lot of
worries about the convergence at infinity. I will not go
through it.
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I mention it only to point up the fact that our existence
theorem doesn’t tell us anything about the behavior of
solutions at infinity or at singular points.
Turning points
Consider the equation
y ′′ − q(x)y = 0.
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The zeroes of q are called “turning points”, or “transfer
points.”
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The behavior of the solutions turns from exponential to
oscillatory at these points.
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The Airy equation is the simplest equation with a turning
point.
Summary
For the equation y ′′ + p(t)y + q(t) = 0 we have
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Wronskian W (u, v ) = uv ′ − vu ′ .
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u and v form a fundamental solution if and only if
W (u, v ) 6= 0.
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Abel’s theorem: W (u, v ) is either identically zero, or never
vanishes.
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Abel’s theorem is proved by showing W ′ + pW = 0.
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Uniqueness is a corollary of Abel’s theorem
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Two classical examples of interest are Bessel’s equation and
Airy’s equation.