Termwise differentiation of the matrix exponential gives
Thus satisfies and . If two solutions existed, multiplying their difference by would give a vector with zero derivative and zero initial value, proving uniqueness.
For the inhomogeneous equation, multiply by the integrating factor :
Integration and multiplication by give the variation-of-constants formula
For the stated matrix,
so
Writing with , one finds
Therefore
or explicitly
Put . The first-order system is
Write
On , is strictly increasing. Moreover,
when . Thus there is one with , in addition to the fixed points at . Explicitly,
Let . Linearization at has eigenvalues satisfying
At zero, , and at , , so both endpoints are saddles. At the interior point,
so is a center. On the full angular interval there is a second center at .
The system has the conserved energy of a conservative planar phase portrait,
For , ,
The phase portrait on the cylinder consists of centers at surrounded by closed periodic-energy curves, with saddles at and the identified point . The saddle-energy contours form the separatrices between librations in the potential wells and trajectories crossing the lower barrier.
If , then and monotonicity gives no interior zero on . The point becomes a center, while remains a saddle. At , the two off-axis centers coalesce with the origin and the linearization there is degenerate.
With and , the chain rule gives
Hence
so the wave equation is equivalent to . Integrating in each variable gives
The initial conditions imply
Solving and integrating yields the D'Alembert formula with initial velocity
If , then both and lie outside the support interval , and every point between them does as well. All three terms therefore vanish. Thus
which is finite propagation speed.
For the finite string, differentiate its energy and use :
The fixed-end conditions hold for every , so differentiation gives . The boundary term vanishes and
For
is an ordinary point when and are analytic there, and a singular point otherwise. A singular point is regular singular when and are analytic there. These are the ordinary-point and regular-singular criteria.
For Kummer's equation, substitute
Equating the coefficient of gives
Taking ,
where is the rising factorial. Thus
Putting and simplifying gives
Therefore
For nonintegral , the powers and at zero are distinct, so these solutions are linearly independent.
When , both solutions tend to . Differentiate their difference with respect to . Writing for derivatives with respect to the second and third arguments,
Hence at one may take
as two linearly independent solutions. Their independence follows from the logarithmic term in the second solution.
Normalization of the probability density function gives
so
The expected value is
The second moment is
Therefore the variance is
For the median ,
so
After tosses, the next total is even if either the current total is even and a tail occurs, or the current total is odd and a head occurs. Therefore
Since , subtracting the fixed point gives
Thus
The formula also covers the endpoint cases and .
Differentiating the Wronskian gives
Thus , the Abel identity.
Also,
If , integration gives the reduction of order formula
For the specified equation, division by gives , so . With , an independent solution is
which is a nonzero multiple of . Hence
The conditions at give and , so
The homogeneous linear recurrence relation
has characteristic polynomial , so
For a particular solution, substitute . The left-hand side becomes
so matching gives and . Hence
The conditions and give and . Therefore
Uniform area density in the unit disk is . The polar coordinates Jacobian gives the joint density
Thus the coordinates are independent and the uniform random point in a disk has
For independent points , the area of triangle is
Now
Independence therefore gives
Conditioned on , the probability that lies inside is its area divided by the disk area . Hence
Four points in general position fail to form a convex quadrilateral exactly when one lies inside the triangle formed by the other three. Each of lies inside the triangle formed by and the other two with probability , and these three events are disjoint. The remaining possibility is that lies inside . By origin in a triangle of three radial random points, this has probability . Therefore

Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.
Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!
We have two killer features:
  1. topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
    Articles of different users are sorted by upvote within each article page. This feature is a bit like:
    • a Wikipedia where each user can have their own version of each article
    • a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
    This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
    Figure 1.
    Screenshot of the "Derivative" topic page
    . View it live at: ourbigbook.com/go/topic/derivative
  2. local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
    This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
    Figure 5. . You can also edit articles on the Web editor without installing anything locally.
    Video 3.
    Edit locally and publish demo
    . Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
  3. https://raw.githubusercontent.com/ourbigbook/ourbigbook-media/master/feature/x/hilbert-space-arrow.png
  4. Infinitely deep tables of contents:
    Figure 6.
    Dynamic article tree with infinitely deep table of contents
    .
    Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
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