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

The mass \(m\) mg of caffeine in a person's blood decreases at a rate proportional to \(m\). At first \(m=120\) and it is decreasing at 15 mg per hour.

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 members of an online forum grows at a rate proportional to \(N\). At first there are 2500 members and the number is increasing at 100 per day.

1. (a) Find \(k\) in \(\dfrac{dN}{dt}=kN\).
2. (b) Find the number after 15 days, to 3 significant figures.

Solve each differential equation.

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

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

An oil slick spreads so that its radius \(r\) m increases at a rate inversely proportional to \(r\). When \(r=25\) the radius is increasing at 0.4 m per hour. \(t\) is the time in hours.

1. (a) Show that \(\dfrac{dr}{dt}=\dfrac{10}{r}\).
2. (b) Find \(r\) in terms of \(t\).
3. (c) Find the time taken for the radius to reach 40 m, to 3 significant figures.

A ball sinks through syrup. Its velocity \(v\) m s⁻¹ after \(t\) seconds satisfies \(\dfrac{dv}{dt}=9-0.45v\), and it starts from rest.

1. (a) Find \(v\) in terms of \(t\).
2. (b) Find the distance it travels in the first 4 seconds, to 3 significant figures.

A spherical ball of ice melts. Its volume decreases at a rate proportional to its surface area. When the volume is 30 cm³ it is decreasing at 4 cm³ per hour. Let \(r\) cm be the radius and \(t\) the time in hours.

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

A bowl of soup at 85 °C is left in a kitchen at 18 °C. Its temperature \(T\) °C after \(t\) minutes decreases at a rate proportional to \(T-18\). After 8 minutes the soup is at 60 °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 30 °C, to 3 significant figures.

A vat holds 500 litres of syrup in which 25 kg of sugar is dissolved. Sugar solution of concentration 0.25 kg per litre flows in at 8 litres per minute. The mixture is stirred and flows out at 8 litres per minute. After \(t\) minutes the vat holds \(m\) kg of sugar.

1. (a) Show that \(\dfrac{dm}{dt}=2-0.016m\).
2. (b) Find \(m\) in terms of \(t\).
3. (c) Find the time taken for the mass to reach 100 kg, to 3 significant figures.
4. (d) State the mass the tank approaches in the long run.

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

1. (a) Show that the general solution can be written as \(2\cos y+\sin x=c\) for some constant \(c\).
2. (b) A particular solution satisfies \(y=\dfrac\pi3\) when \(x=\dfrac\pi2\), for \(0\le x\le\pi\) and \(-\pi\le y\le\pi\). Find the two possible values of \(y\) when \(x=\dfrac\pi6\), to 3 significant figures.