Setting up and integrating Problem set A

Write a differential equation for each situation. Define your variables first.

1. (a) The mass of mould on a wall grows at a rate proportional to its mass.
2. (b) The height of a bamboo shoot increases at a rate proportional to the cube root of its height.

Write a differential equation for each situation. Define your variables first.

1. (a) The thickness of ice on a pond increases at a rate inversely proportional to the thickness.
2. (b) The radius of a growing bubble increases at a rate inversely proportional to the cube of the radius.

Write a differential equation for each situation. Define your variables first.

1. (a) The number of unsold tickets decreases at a rate proportional to the square root of that number.
2. (b) The speed of a crate sliding down a ramp decreases at a rate inversely proportional to the square of its speed.

Find the general solution of each differential equation.

1. (a) \(\dfrac{dy}{dx}=2x^{3} + \frac{6}{\sqrt{x}}\)
2. (b) \(\dfrac{dy}{dt}=5e^{-\frac{t}{2}}\)
3. (c) \(\dfrac{dy}{dt}=2 + \frac{8}{t}\), \(t>0\)
4. (d) \(\dfrac{ds}{dt}=4 \cos(2t)\)

A town has 5000 residents. After \(t\) hours, \(S\) of them have seen a viral clip. The rate at which \(S\) increases is proportional to the number who have seen the clip and to the number who have not. Write a differential equation.

A hashtag spreads through a group of 3000 people. After \(t\) hours, \(H\) of them have used it. The rate at which \(H\) increases is proportional to the number who have used it and to the number who have not, and is inversely proportional to \(t\). Write a differential equation.

A cyclist's velocity is \(v=\dfrac{ds}{dt}=6e^{-0.2t}\) m s⁻¹, where \(s\) is the distance from the start in metres and \(t\) is the time in seconds.

1. (a) Find the general solution for \(s\) in terms of \(t\).
2. (b) Given that the cyclist starts at \(s=0\), find the distance after 10 seconds.

Rearrange each equation to make the derivative the subject, then find the general solution.

1. (a) \(\dfrac{dF}{dm}+\frac{2}{m^{2}}=5\), \(m>0\)
2. (b) \(\dfrac{dG}{dt}+8e^{-2t}=1\)