Wednesday 27 March 2013

Series Solutions: Airy's Equation


The general form of a homogeneous second order linear differential equation looks as follows:

y''+p(ty'+q(ty=0.



The series solutions method is used primarily, when the coefficients p(t) or q(t) are non-constant.

One of the easiest examples of such a case is Airy's Equation

y''-t y=0,


which is used in physics to model the defraction of light.
We want to find power series solutions for this second-order linear differential equation.

The generic form of a power series is

\begin{displaymath}y(t)=\sum_{n=0}^\infty a_n t^n.\end{displaymath}


We have to determine the right choice for the coefficients (an).
As in other techniques for solving differential equations, once we have a "guess" for the solutions, we plug it into the differential equation. Recall that

\begin{displaymath}y''(t)=\sum_{n=2}^\infty n(n-1) a_n t^{n-2}.\end{displaymath}



Plugging this information into the differential equation we obtain:

\begin{displaymath}\sum_{n=2}^\infty n(n-1) a_n t^{n-2}-t\sum_{n=0}^\infty a_n t^n=0, \end{displaymath}


or equivalently

\begin{displaymath}\sum_{n=2}^\infty n(n-1) a_n t^{n-2}-\sum_{n=0}^\infty a_n t^{n+1}=0. \end{displaymath}



Our next goal is to simplify this expression such that (basically) only one summation sign "$\sum$" remains. The obstacle we encounter is that the powers of both sums are different, tn-2 for the first sum and tn+1 for the second sum. We make them the same by shifting the index of the first sum up by 2 units and the index of the second sum down by one unit to obtain



\begin{displaymath}\sum_{n=0}^\infty (n+2)(n+1) a_{n+2} t^{n}-\sum_{n=1}^\infty a_{n-1} t^{n}=0.\end{displaymath}



Now we run into the next problem: the second sum starts at n=1, while the first sum has one more term and starts at n=0. We split off the 0th term of the first sum:

\begin{displaymath}\sum_{n=0}^\infty (n+2)(n+1) a_{n+2} t^{n}=2\cdot 1\cdot a_2+
\sum_{n=1}^\infty (n+2)(n+1) a_{n+2} t^{n}.\end{displaymath}



Now we can combine the two sums as follows:

\begin{displaymath}2 a_2 +\sum_{n=1}^\infty \left( \phantom{\int}(n+2)(n+1) a_{n+2} t^{n}- a_{n-1} t^n\phantom{\int}\right)=0,\end{displaymath}


and factor out tn:

\begin{displaymath}2 a_2+\sum_{n=1}^\infty \left( \phantom{\int}(n+2)(n+1) a_{n+2} - a_{n-1} \phantom{\int}\right)t^n=0.\end{displaymath}




The power series on the left is identically equal to zero, consequently all of its coefficients are equal to 0:

\begin{displaymath}\left\{\begin{array}{cl}2a_2&=0\\ (n+2)(n+1) a_{n+2} - a_{n-1} &=0 \mbox{ for all } n=1,2,3,\ldots\end{array}\right.\end{displaymath}



We can slightly rewrite as

\begin{displaymath}\left\{\begin{array}{ll}a_2&=0\\ a_{n+2}& =\displaystyle{\fra...
...{(n+1)(n+2)}} \mbox{ for all } n=1,2,3,\ldots\end{array}\right.\end{displaymath}


These equations are known as the "recurrence relations" of the differential equations. The recurrence relations permit us to compute all coefficients in terms of a0 and a1.
We already know from the 0th recurrence relation that a2=0. Let's compute a3 by reading off the recurrence relation for n=1:

\begin{displaymath}a_3=\frac{a_0}{2\cdot 3}.\end{displaymath}


Let us continue:

\begin{eqnarray*}a_4&=&\frac{a_1}{3\cdot 4}\\
a_5&=&\frac{a_2}{4\cdot 5}=0\\
a...
...&\frac{a_6}{8\cdot 9}=\frac{a_0}{(2\cdot 3)(5\cdot 6)(8\cdot 9)}
\end{eqnarray*}



The hardest part, as usual, is to recognize the patterns evolving; in this case we have to consider three cases:

1. All the terms $a_2,a_5,a_8,\ldots$ are equal to zero. We can write this in compact form as

\begin{displaymath}a_{3k+2}=0\mbox{ for all }k=0,1,2,3,\ldots\end{displaymath}



2. All the terms $a_3,a_6,a_9,\ldots$ are multiples of a0. We can be more precise:

\begin{displaymath}a_{3k}=\frac{1}{(2\cdot 3)(5\cdot 6)\cdots((3k-1)\cdot (3k))}\cdot a_0
\mbox{ for all }k=1,2,3,\ldots\end{displaymath}


(Plug in k=1,2,3,4 to check that this works!)
3. All the terms $a_4,a_7,a_{10},\ldots$ are multiples of a1. We can be more precise:

\begin{displaymath}a_{3k+1}=\frac{1}{(3\cdot 4)(6\cdot 7)\cdots((3k)\cdot (3k+1))}\cdot a_1
\mbox{ for all }k=1,2,3,\ldots\end{displaymath}


(Plug in k=1,2,3,4 to check that this works!)
Thus the general form of the solutions to Airy's Equation is given by

\begin{eqnarray*}y(t)&=&a_0\left(1+\sum_{k=1}^\infty \frac{t^{3k}}{(2\cdot 3)(5\...
...{t^{3k+1}}{(3\cdot 4)(6\cdot 7)\cdots((3k)\cdot (3k+1))}\right).
\end{eqnarray*}



Note that, as always, y(0)=a0 and y'(0)=a1. Thus it is trivial to determine a0 and a1 when you want to solve an initial value problem.

In particular

\begin{displaymath}y_1(t)= 1+\sum_{k=1}^\infty \frac{t^{3k}}{(2\cdot 3)(5\cdot 6)\cdots((3k-1)\cdot (3k))}\end{displaymath}


and

\begin{displaymath}y_2(t)=t+\sum_{k=1}^\infty\frac{t^{3k+1}}{(3\cdot 4)(6\cdot 7)\cdots((3k)\cdot (3k+1))}\end{displaymath}


form a fundamental system of solutions for Airy's Differential Equation.
Below you see a picture of these two solutions. Note that for negative t, the solutions behave somewhat like the oscillating solutions of y''+y=0, while for positive t, they behave somewhat like the exponential solutions of the differential equation y''-y=0.



Reduction of Order Technique


This technique is very important since it helps one to find a second solution independent from a known one. , in order to find the general solution to y'' + p(x)y' + q(x)y = 0, we need only to find one (non-zero) solution, tex2html_wrap_inline143 .
Let tex2html_wrap_inline143 be a non-zero solution of
displaymath147
Then, a second solution tex2html_wrap_inline149 independent of tex2html_wrap_inline143 can be found as
displaymath135
Easy calculations give
displaymath136,
where C is an arbitrary non-zero constant. Since we are looking for a second solution one may take C=1, to get
displaymath157
Remember that this formula saves time. But, if you forget it you will have to plug tex2html_wrap_inline159 into the equation to determine v(x) which may lead to mistakes !
The general solution is then given by
displaymath137

Example: Find the general solution to the Legendre equation
displaymath163,
using the fact that tex2html_wrap_inline165 is a solution.
Solution: It is easy to check that indeed tex2html_wrap_inline165 is a solution. First, we need to rewrite the equation in the explicit form
displaymath169
We may try to find a second solution tex2html_wrap_inline171 by plugging it into the equation. We leave it to the reader to do that! Instead let us use the formula
displaymath173
Techniques of integration (of rational functions) give
displaymath175,
which gives
displaymath177
The general solution is then given by
displaymath179

Vertical Tangents and Cusps


In the definition of the slope, vertical lines were excluded. It is customary not to assign a slope to these lines. This is true as long as we assume that a slope is a number. But from a purely geometric point of view, a curve may have a vertical tangent. Think of a circle (with two vertical tangent lines). We still have an equation, namely x=c, but it is not of the form y = ax+b. In fact, such tangent lines have an infinite slope. To be precise we will say:
The graph of a function f(x) has a vertical tangent at the point (x0,f(x0)) if and only if 

\begin{displaymath}f'(x) \rightarrow +\infty\;\;\mbox{or}\;\; f'(x) \rightarrow -\infty \;\;\mbox{as}\;\; x \rightarrow x_0\;.\end{displaymath}


Example. Consider the function

\begin{displaymath}f(x) = \sqrt[5]{2-x}\end{displaymath}


We have

\begin{displaymath}f'(x) = -\frac{1}{5} \frac{1}{(2-x)^{4/5}}\;\cdot\end{displaymath}


Clearly, f'(2) does not exist. In fact we have

\begin{displaymath}\lim_{x \rightarrow 2} f'(x) = -\infty \cdot\end{displaymath}


So the graph of f(x) has a vertical tangent at (2,0). The equation of this line is x=2.


In this example, the limit of f'(x) when $x \rightarrow 2$ is the same whether we get closer to 2 from the left or from the right. In many examples, that is not the case.
Example. Consider the function

\begin{displaymath}f(x) = x^{2/3}\;.\end{displaymath}


We have

\begin{displaymath}f'(x) = \frac{2}{3} \frac{1}{x^{1/3}} \cdot\end{displaymath}


So we have

\begin{displaymath}\lim_{x \rightarrow 0+} f'(x) = +\infty \;\;\mbox{and}\;\; \lim_{x \rightarrow 0-} f'(x) = -\infty
\cdot\end{displaymath}





It is clear that the graph of this function becomes vertical and then virtually doubles back on itself. Such pattern signals the presence of what is known as a vertical cusp. In general we say that the graph of f(x) has a vertical cusp at x0,f(x0)) iff

\begin{displaymath}f'(x) \rightarrow +\infty \;\;\mbox{as}\;\; x \rightarrow x_0...
...
f'(x) \rightarrow -\infty \;\;\mbox{as}\;\; x \rightarrow x_0-\end{displaymath}


or

\begin{displaymath}f'(x) \rightarrow -\infty \;\;\mbox{as}\;\; x \rightarrow x_0...
...(x) \rightarrow +\infty \;\;\mbox{as}\;\; x \rightarrow
x_0-\;.\end{displaymath}


In both cases, f'(x0) becomes infinite. A graph may also exhibit a behavior similar to a cusp without having infinite slopes:
Example. Consider the function

f(x) = |x3 - 8|.


Clearly we have

\begin{displaymath}f(x) = \left\{\begin{array}{cll}
-(x^3 - 2) & \mbox{if $x \leq 2$}\\
x^3 - 2 & \mbox{if $x \geq 2$}.
\end{array}\right.\end{displaymath}


Hence

\begin{displaymath}f'(x) = \left\{\begin{array}{rll}
- 3x^2 & \mbox{if $x < 2$}\\
3x^2 & \mbox{if $x > 2$}.
\end{array}\right.\end{displaymath}


Direct calculations show that f'(2) does not exist. In fact, we have left and right derivatives with

\begin{displaymath}f'_-(2) = -12 \;\;\mbox{and}\;\; f'_+(2) = 12\;.\end{displaymath}





So there is no vertical tangent and no vertical cusp at x=2. In fact, the phenomenon this function shows at x=2 is usually called a corner.

https://www.youtube.com/TarunGehlot