Differentiation Problem

[icinimod]

Suppose that y is a function of x whose graph goes through the point (ln9,24). Further suppose that (dy/dx) = .5y. Determine a formula for y.

I don't even know where to start on a problem like this. Any help would be greatly appreciated.

 

[BigGlenntheHeavy]

$\displaystyle Given: \ f(x) \ = \ y, \ f[(ln(9)] \ = \ 24, \ \frac{dy}{dx} \ = \ .5y, \ Find \ f(x).$

$\displaystyle \frac{dy}{dx} \ = \ .5y \ \implies \ dy \ = \ .5ydx, \ ergo,$

$\displaystyle \int dy \ = \ \int.5ydx, \ \int\frac{dy}{y} \ = \ \int.5dx$

$\displaystyle ln|y| \ = \ .5x \ + \ C, \ y \ = \ Ae^{.5x}, \ A \ = \ \pm e^{C}, \ or \ A \ = \ 0$

$\displaystyle f[ln(9)] \ = \ 24 \ = \ Ae^{ln(3)} \ = \ 3A, \ A \ = \ 8, \ hence \ f(x) \ = \ 8e^{.5x}$

 

[icinimod]

Thanks for the quick reply Big G.