Pure Mathematics Practice: Differential Equations

Original practice material

These four structured questions total 32 marks. They practise first-order separable equations and model interpretation suitable for Paper 1 or Paper 2 Section A; they are not a complete paper.

Answers: Differential Equations Answers.

Question 1: Basic Separation

Solve

$$\frac{dy}{dx}=3x^2(1+y),y(0)=2.$$

[7]

Question 2: Exponential Decay Model

A quantity $N$ satisfies

$$\frac{dN}{dt}=-0.04N,N(0)=500.$$

(a) Find $N$ in terms of $t$. [4]

(b) Find the time when $N=300$, giving your answer to 3 significant figures. [3]

[Total: 7]

Question 3: Logistic-Type Separation

Solve

$$\frac{dy}{dx}=y(4-y),y(0)=1,$$

leaving your answer in explicit form. [9]

Question 4: A Draining-Tank Depth Model

A tank has constant horizontal cross-sectional area. While liquid remains in the tank, its depth $h$ metres is modelled by

$$\frac{dh}{dt}=-k\sqrt{h},k>0,$$

where $t$ is measured in seconds. Given that $h=16$ when $t=0$ and $h=9$ when $t=10$, find $k$ and the time at which the model predicts that the tank becomes empty. State the time interval on which your expression for $h$ is physically valid. [9]