I've got a problem and i should solve it using differential equation.I don't know how to write the equation and start.
A person is trying to fill a bathtub with water. Water is flowing into the bathtub from the tap at a constant rate of k litres/sec. However, there is a hole in the bottom of the bathtub and water is flowing out of the bathtub at a rate proportional to the square of the volume of water present in the bathtub. If V(t) is the volume of water (in litres) present in the bathtub at time t (in seconds) and the bathtub initially contains V(0) litres of water.
How can i write down the differential equation regarding this problem. not solve just writing the equation?
Best Answer
The conservation equations dictate that the rate of change in volume is given by:
$$ \frac{\mathrm{d}V(t)}{\mathrm{d}t} = \dot{V}_{in}-\dot{V}_{out}, $$
where $\dot{V}_{in}$ and $\dot{V}_{out}$ are the corresponding inflow and outflow. Note that the units of both magnitudes are $[V]/[t]$, which in your case are $L/s$ so we have $\dot{V}_{in} = k$ and $\dot{V}_{out} = \alpha V^2$, where $\alpha$ is a constant of proportionality (measured in $s/L$). Thus, we come up with the following nonlinear first order ODE:
$$V' + \alpha V^2 = k, \quad V(0) = V_0.$$
Can you solve this Bernouilli differential equation?
Cheers!