Differential Equations

Show that

Given that $h = 0$ when $t = 0$, find the time taken for the depth of the water in the tank to reach $20\text{ cm}$.

Explain why, after $t$ years, $\frac{dx}{dt} = kx(1 - x)$, where $k$ is a constant.

When the disease is first detected, one quarter of the trees are affected.
Two years later, one third of the trees are affected.

Solve the differential equation to find the number of years from the time when the disease is first detected until the time when three quarters of the trees are affected. Give your answer correct to the nearest year.

Show that $x$ and $t$ satisfy the differential equation

Using partial fractions, solve the differential equation and obtain an expression for $t$ in terms of a single logarithm involving $x$.

Show that $x$ and $t$ satisfy the differential equation

Solve the differential equation, obtaining an expression for $t$ in terms of $x$.

Show that $\frac{dh}{dt} = -\lambda \sqrt{h}$, where $\lambda$ is a positive constant.

At time $t = 0$ the tap is opened. It is given that $h = 4$ when $t = 0$ and that $h = 2.25$ when $t = 20$.

Solve the differential equation to obtain an expression for $t$ in terms of $h$, and hence find the time taken to empty the tank.