marginal revenue function:
dr/dq=(50-4q)e^(-r/5)
find the demand equation
i dont even know where to start for this one, but its related to differential equations
any help is appreciated!
This is a seperable differential equation.
$\displaystyle \frac{dr}{dq}=(50-4q)e^{-r/5}$ moving around some stuff we get..
$\displaystyle e^{r/5}dr=(50-4q)dq$ so now we integrate both sides.
$\displaystyle \int e^{r/5}dr=\int (50-4q)dq \iff 5e^{r/5}=-2q^2+50q+C$
If we want we can solve the equation for r
$\displaystyle e^{r/5}=-\frac{2}{5}q^2+10q+\frac{C}{5}$ taking the ln of both sides.
$\displaystyle \frac{r}{5}=ln \left[-\frac{2}{5}q^2+10q+\frac{C}{5} \right]$
$\displaystyle r=5ln \left[-\frac{2}{5}q^2+10q+\frac{C}{5} \right]$