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248

Calculus Made Easy

equation represents the propagation of a wave (of any form) at a uniform speed along the $x$ direction.

If the differential equation had been written

$m{\frac {d^{2}y}{dt^{2}}}=k{\frac {d^{2}y}{dx^{2}}}$ ,

the solution would have been the same, but the velocity of propagation would have had the value

$a={\sqrt {\frac {k}{m}}}$

.

You have now been personally conducted over the frontiers into the enchanted land. And in order that you may have a handy reference to the principal results, the author, in bidding you farewell, begs to present you with a passport in the shape of a convenient collection of standard forms (see pp. 252, 253). In the middle column are set down a number of the functions which most commonly occur. The results of differentiating them are set down on the left; the results of integrating them are set down on the right. May you find them useful!

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