In the diagram, the tangent to a curve at the point P with coordinates (x, y) meets the x-axis at T. The point N is the foot of the perpendicular from P to the x-axis. The curve is such that, for all values of x, the gradient of the curve is positive and T N=2.

(a)

Show that the differential equation satisfied by x and y is $\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{1}{2} y$ .

[ 1 ]

(b)

The point with coordinates (4,3) lies on the curve.

Solve the differential equation to obtain the equation of the curve, expressing y in terms of x.

[ 5 ]

[Maximum number: 7]

The number of insects in a population t weeks after the start of observations is denoted by N. The population is decreasing at a rate proportional to $N \mathrm{e}^{-0.02 t}$ . The variables N and t are treated as continuous, and it is given that when t=0, N=1000 and $\frac{\mathrm{d} N}{\mathrm{~d} t}=-10$ .

(a)

Show that N and t satisfy the differential equation

\frac{\mathrm{d} N}{\mathrm{~d} t}=-0.01 \mathrm{e}^{-0.02 t} N .

(b)

Solve the differential equation and find the value of t when N=800.

[ 6 ]

(c)

State what happens to the value of N as t becomes large.

[ 1 ]

[Maximum number: 7]

The variables x and y satisfy the differential equation

(1-\cos x) \frac{\mathrm{d} y}{\mathrm{~d} x}=y \sin x

It is given that y=4 when $x=\pi$ .

(a)

Solve the differential equation, obtaining an expression for y in terms of x.

[ 6 ]

(b)

Sketch the graph of y against x for $0<x<2 \pi$ .

[ 1 ]

(a)

The variables x and $\theta$ satisfy the differential equation

x \tan \theta \frac{\mathrm{~d} x}{\mathrm{~d} \theta}+\operatorname{cosec}^{2} \theta=0

for $0<\theta<\frac{1}{2} \pi$ and x>0. It is given that x=4 when $\theta=\frac{1}{6} \pi$ . Solve the differential equation, obtaining an expression for x in terms of $\theta$ .

[ 6 ]

[Maximum number: 6]

The variables x and y are related by the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{6 y \mathrm{e}^{3 x}}{2+\mathrm{e}^{3 x}}

Given that y=36 when x=0, find an expression for y in terms of x.

[Maximum number: 7]

The variables x and $\theta$ are related by the differential equation

\sin 2 \theta \frac{\mathrm{~d} x}{\mathrm{~d} \theta}=(x+1) \cos 2 \theta

where $0<\theta<\frac{1}{2} \pi$ . When $\theta=\frac{1}{12} \pi, x=0$ . Solve the differential equation, obtaining an expression for x in terms of $\theta$ , and simplifying your answer as far as possible.

[Maximum number: 7]

Given that x=1 when t=0, solve the differential equation

\frac{\mathrm{d} x}{\mathrm{~d} t}=\frac{1}{x}-\frac{x}{4},

obtaining an expression for $x^{2}$ in terms of t.

[Maximum number: 7]

The variables x and y satisfy the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x y}{1+x^{2}}

and y=2 when x=0.
Solve the differential equation, obtaining a simplified expression for y in terms of x.

[Maximum number: 1]

During an experiment, the number of organisms present at time t days is denoted by N, where N is treated as a continuous variable. It is given that

\frac{\mathrm{d} N}{\mathrm{~d} t}=1.2 \mathrm{e}^{-0.02 t} N^{0.5}

When t=0, the number of organisms present is 100 .

(a)

Find an expression for N in terms of t.

(b)

State what happens to the number of organisms present after a long time.

[ 1 ]

[Maximum number: 7]

The variables x and $\theta$ are related by the differential equation

\sin 2 \theta \frac{\mathrm{~d} x}{\mathrm{~d} \theta}=(x+1) \cos 2 \theta

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