Page:Calculus Made Easy.pdf/267

 Example 8.

It was seen (p. 177) that this equation was derived from the original

where $$F$$ and $$f$$ were any arbitrary functions of $$t$$.

Another way of dealing with it is to transform it by a change of variables into

where $$u=x+at$$, and $$v=x-at$$, leading to the same general solution. If we consider a case in which $$F$$ vanishes, then we have simply

and this merely states that, at the time $$t=0$$, $$y$$ is a particular function of $$x$$, and may be looked upon as denoting that the curve of the relation of $$y$$ to $$x$$ has a particular shape. Then any change in the value of $$t$$ is equivalent simply to an alteration in the origin from which $$x$$ is reckoned. That is to say, it indicates that, the form of the function being conserved, it is propagated along the $$x$$ direction with a uniform velocity $$a$$; so that whatever the value of the ordinate $$y$$ at any particular time $$t_0$$ at any particular point $$x_0$$, the same value of $$y$$ will appear at the subsequent time $$t_1$$ at a point further along, the abscissa of which is $$x_0+a(t_1-t_0)$$. In this case the simplified