Separable equations in context: growth, decay and harder forms Problem set A

A sample of a radioactive substance decays at a rate proportional to its mass \(m\) g. At first the mass is 60 g and it is decreasing at 7.2 g per day.

1. (a) Find the constant \(k\) in \(\dfrac{dm}{dt}=-km\).
2. (b) Find \(m\) in terms of \(t\).
3. (c) Find how long it takes for the mass to halve, to 3 significant figures.

The number \(N\) of yeast cells in a vat grows at a rate proportional to \(N\). At first there are 800 cells and the number is increasing at 56 cells per hour.

1. (a) Find \(k\) in \(\dfrac{dN}{dt}=kN\).
2. (b) Find the number after 6 hours, to 3 significant figures.

Solve each differential equation.

1. (a) Find the general solution of \(\dfrac{dy}{dx}=e^{3x+y}\).
2. (b) Find the particular solution of \(\dfrac{dy}{dx}=5e^{x-4y}\), \(y=0\) when \(x=0\). Give \(y\) in terms of \(x\).

Given that \(y\dfrac{dy}{dx}=3e^{-x}\) and that \(y=-2\) when \(x=0\), find \(y\) in terms of \(x\).

Water flows into a tank so that the depth \(h\) cm increases at a rate inversely proportional to \(h\). When \(h=40\) the depth is increasing at 0.5 cm per minute. \(t\) is the time in minutes.

1. (a) Show that \(\dfrac{dh}{dt}=\dfrac{20}{h}\).
2. (b) Find \(h\) in terms of \(t\).
3. (c) Find the time taken for the depth to reach 60 cm, to 3 significant figures.

An object falls through air. Its velocity \(v\) m s⁻¹ after \(t\) seconds satisfies \(\dfrac{dv}{dt}=14-0.35v\), and it starts from rest.

1. (a) Find \(v\) in terms of \(t\).
2. (b) Find the distance it travels in the first 3 seconds, to 3 significant figures.

A spherical mint dissolves. Its volume decreases at a rate proportional to its surface area. When the volume is 12 cm³ it is decreasing at 1.5 cm³ per minute. Let \(r\) cm be the radius and \(t\) the time in minutes.

1. (a) Find \(\dfrac{dr}{dt}\).
2. (b) Find the time taken to dissolve completely.

A metal block is heated to 150 °C and placed in a room at 22 °C. Its temperature \(T\) °C after \(t\) minutes decreases at a rate proportional to \(T-22\). After 6 minutes the block is at 100 °C.

1. (a) Write down a differential equation for \(T\), using a constant \(k\).
2. (b) Find \(T\) in terms of \(t\).
3. (c) Find how long it takes to cool to 40 °C, to 3 significant figures.

A tank holds 300 litres of water in which 10 kg of salt is dissolved. Salt solution of concentration 0.1 kg per litre flows in at 6 litres per minute. The mixture is stirred and flows out at 6 litres per minute. After \(t\) minutes the tank holds \(m\) kg of salt.

1. (a) Show that \(\dfrac{dm}{dt}=0.6-0.02m\).
2. (b) Find \(m\) in terms of \(t\).
3. (c) Find the time taken for the mass to reach 20 kg, to 3 significant figures.
4. (d) State the mass the tank approaches in the long run.

Given that \(\dfrac{dy}{dx}=\dfrac{2\cos x}{\sin y}\):

1. (a) Show that the general solution can be written as \(\cos y+2\sin x=c\) for some constant \(c\).
2. (b) A particular solution satisfies \(y=\dfrac{2\pi}{3}\) when \(x=\dfrac\pi2\), for \(0\le x\le\pi\) and \(-\pi\le y\le\pi\). Find the two possible values of \(y\) when \(x=\dfrac{5\pi}{6}\), to 3 significant figures.