7.10 Constructing differential equations — Practise with help

Work each question closed book first. Read the sentence twice: once to find the rate, and once to find the sign.

Warm-up: remember

Write " $y$ is proportional to $x^2$ " as an equation.
Make $k$ the subject of $5 = 2k$ .
Differentiate $y = Ae^{3x}$ with respect to $x$ .

Answers

$y = kx^2$ , where $k$ is a constant.
$k = \dfrac{5}{2} = 2.5$ .
$\dfrac{dy}{dx} = 3Ae^{3x}$ .

If question 1 was hard, revise proportionality in Chapter 2.

Practise with help

1. A population $P$ of insects grows at a rate proportional to $P$ . Write a differential equation relating $\dfrac{dP}{dt}$ to $P$ .

Hint

"Proportional to $P$ " introduces a constant $k$ . The population grows, so the sign is positive.

Next step

Write the rate on the left and the proportional expression on the right: $\dfrac{dP}{dt}$ equals a constant times $P$ .

Full working

\frac{dP}{dt} = kP, \qquad k > 0.

The larger the population, the faster it grows, because both sides grow with $P$ .

2. A radioactive sample of mass $m$ grams decays at a rate proportional to its mass. Write a differential equation for $m$ after $t$ years.

Hint

Decay means the mass is falling, so the rate is negative. "Proportional to its mass" introduces a constant $k$ .

Next step

Write $\dfrac{dm}{dt}$ equal to minus a constant times $m$ .

Full working

\frac{dm}{dt} = -km, \qquad k > 0.

The minus sign says the mass decreases. The constant $k$ is the decay constant, and a larger $k$ means faster decay.

3. A metal block cools in air at $15^\circ$ C. Its temperature $\theta$ falls at a rate proportional to the difference between $\theta$ and the air temperature. Write a differential equation for $\theta$ after $t$ minutes.

Hint

The temperature difference is $\theta - 15$ . The block cools, so the rate is negative while $\theta > 15$ .

Next step

Write $\dfrac{d\theta}{dt} = -k(\theta - 15)$ .

Full working

\frac{d\theta}{dt} = -k(\theta - 15), \qquad k > 0.

When $\theta > 15$ the right side is negative, so the temperature falls. As $\theta$ approaches $15$ the right side approaches zero, so the cooling slows down.

4. A car accelerates at a constant rate $a$ . Its velocity is $v$ after $t$ seconds. Write a differential equation for $v$ .

Hint

Acceleration is the rate of change of velocity.

Next step

Write the rate of change of $v$ equal to the constant $a$ .

Full working

\frac{dv}{dt} = a.

A constant acceleration means the velocity changes at a steady rate, so the graph of $v$ against $t$ is a straight line of gradient $a$ .

5. The curve $y = A\sin 2x$ contains a constant $A$ . Form a differential equation that does not contain $A$ .

Hint

Differentiate twice. The second derivative brings back $-\sin 2x$ , which is proportional to $y$ .

Next step

$\dfrac{dy}{dx} = 2A\cos 2x$ , so $\dfrac{d^2y}{dx^2} = -4A\sin 2x$ . Now replace $A\sin 2x$ with $y$ .

Full working

\frac{dy}{dx} = 2A\cos 2x, \qquad \frac{d^2y}{dx^2} = -4A\sin 2x = -4y.

The differential equation is $\dfrac{d^2y}{dx^2} = -4y$ . Every curve in the family satisfies it, whatever the value of $A$ .

6. The curve $y = x^2 + C$ contains a constant $C$ . Form a differential equation that does not contain $C$ .

Hint

Differentiate once and see whether the constant survives.

Next step

$\dfrac{dy}{dx} = 2x$ . The constant $C$ has already gone.

Full working

\frac{dy}{dx} = 2x.

Every curve $y = x^2 + C$ has the same gradient at a given $x$ , because adding a constant moves the curve up or down without changing its slope.