Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/175



Therefore

and so on for higher derivatives. This transformation is called changing the independent variable from x to t. It is usually better to work out examples by the methods illustrated above rather than by using the formulas deduced.

Change the independent variable from x to t in the equation.


 * Solution.


 * {| style="width:100%"

= e^{-t} \frac{d}{dt} \left( \frac{dy}{dt} \right) \frac{dt}{dx} - \frac{dy}{dt} e^{-t} \frac{dt}{dx}.$$
 * align="center"|$$\frac{dx}{dt} = e^t\ $$, therefore
 * style="width:3%"|
 * align="center"|$$\frac{dt}{dx}= e^{-t}\ $$.
 * Also
 * align="center"|$$\frac{dy}{dx}= \frac{dy}{dt} \frac{dt}{dx}$$; therefore
 * align="center"|$$\frac{dy}{dx} = e^{-t} \frac{dy}{dt}$$
 * Also
 * align="center"|$$\frac{d^2 y}{dx^2} = e^{-t} \frac{d}{dx} \left( \frac{dy}{dt} \right) - \frac{dy}{dt} e^{-t} \frac{dt}{dx}
 * align="center"|$$\frac{dy}{dx} = e^{-t} \frac{dy}{dt}$$
 * Also
 * align="center"|$$\frac{d^2 y}{dx^2} = e^{-t} \frac{d}{dx} \left( \frac{dy}{dt} \right) - \frac{dy}{dt} e^{-t} \frac{dt}{dx}
 * Also
 * align="center"|$$\frac{d^2 y}{dx^2} = e^{-t} \frac{d}{dx} \left( \frac{dy}{dt} \right) - \frac{dy}{dt} e^{-t} \frac{dt}{dx}
 * align="center"|$$\frac{d^2 y}{dx^2} = e^{-t} \frac{d}{dx} \left( \frac{dy}{dt} \right) - \frac{dy}{dt} e^{-t} \frac{dt}{dx}
 * }


 * Substituting in the last result from (E),


 * {| style="width:100%"


 * style="width:3%"|
 * align="center"|$$\frac{d^2 y}{dx^2}= e^{-2t} \frac{d^2 y}{dt^2} - \frac{dy}{dt} e^{-2t};$$
 * }


 * Substituting (D), (F), (G) in (C),

$$e^{2t} \left( e^{-2t} \frac{d^2 y}{dt^2} - \frac{dy}{dt} e^{-2t} \right) + e^t \left( e^{-t} \frac{dy}{dt} \right) + y = 0;$$


 * and reducing, we get

$$\frac{d^2 y}{dt^2} + y = 0\ $$ Ans.

Since the formulas deduced in the Differential Calculus generally involve derivatives of y with respect to x, such formulas as (A) and (B) are especially useful when the parametric equations of a curve are given. Such examples were given in §66, and many others will be employed in what follows.