Page:Calculus Made Easy.pdf/268

 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 $m\frac{d^2y}{dt^2}=k\frac{d^2y}{dx^2}$, the solution would have been the same, but the velocity of propagation would have had the value $a=\sqrt {\frac{k}{m}}$|undefined.

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!