Page:Calculus Made Easy.pdf/258

 If the condition is laid down that $$y=0$$ when $$x=0$$ we can find $$C$$; for then the exponential becomes $$=1$$; and we have

or

Putting in this value, the solution becomes

But further, if $$x$$ grows indefinitely, $$y$$ will grow to a maximum; for when $$x=\infty$$, the exponential $$=0$$, giving $$y_{\text{max.}} = \dfrac{g}{a}$$. Substituting this, we get finally

This result is also of importance in physical science.

Example 3. Let $$\quad\quad\quad\quad ay+b\frac{dy}{dt} = g \cdot \sin 2\pi nt$$.

We shall find this much less tractable than the preceding. First divide through by $$b$$.

Now, as it stands, the left side is not integrable. But it can be made so by the artifice–and this