\(\displaystyle Given: \ f(x) \ = \ y, \ f[(ln(9)] \ = \ 24, \ \frac{dy}{dx} \ = \ .5y, \ Find \ f(x).\)
\(\displaystyle \frac{dy}{dx} \ = \ .5y \ \implies \ dy \ = \ .5ydx, \ ergo,\)
\(\displaystyle \int dy \ = \ \int.5ydx, \ \int\frac{dy}{y} \ = \ \int.5dx\)
\(\displaystyle ln|y| \ = \ .5x \ + \ C, \ y \ = \ Ae^{.5x}, \ A \ = \ \pm e^{C}, \ or \ A \ = \ 0\)
\(\displaystyle f[ln(9)] \ = \ 24 \ = \ Ae^{ln(3)} \ = \ 3A, \ A \ = \ 8, \ hence \ f(x) \ = \ 8e^{.5x}\)
