A-Level CAIE Mathematics A2 3.8 Differential equations Question Bank

A particle P of mass 0.2 kg is released from rest at a point O on a smooth horizontal surface. A horizontal force of magnitude $t \mathrm{e}^{-v} \mathrm{~N}$ directed away from O acts on P, where $v \mathrm{~m} \mathrm{~s}^{-1}$ is the velocity of P at time $t \mathrm{~s}$ after release. Find the velocity of P when t=2.

A particle P moves in a straight line and passes through a point O of the line with velocity $2 \mathrm{~m} \mathrm{~s}^{-1}$. At time $t \mathrm{~s}$ after passing through O, the velocity of P is $v \mathrm{~m} \mathrm{~s}^{-1}$ and the acceleration of P is given by $\mathrm{e}^{-0.5 v} \mathrm{~m} \mathrm{~s}^{-2}$. Calculate the velocity of P when t=1.2.

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 particle P of mass 0.4 kg is released from rest at a point O on a smooth plane inclined at $30^{\circ}$ to the horizontal. P moves down the line of greatest slope through O. The velocity of P is $v \mathrm{~m} \mathrm{~s}^{-1}$ when its displacement from O is $x \mathrm{~m}$. A retarding force of magnitude $0.2 v^{2} \mathrm{~N}$ acts on P in the direction P O.

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 $\frac{\mathrm{d} N}{\mathrm{~d} t}=-10$.

The variables x and y satisfy the differential equation

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

A small ball of mass $m \mathrm{~kg}$ is projected vertically upwards with speed $14 \mathrm{~m} \mathrm{~s}^{-1}$. The ball has velocity $v \mathrm{~m} \mathrm{~s}^{-1}$ upwards when it is $x \mathrm{~m}$ above the point of projection. A resisting force of magnitude $0.02 m v \mathrm{~N}$ acts on the ball during its upward motion.

A smooth horizontal surface has two fixed points O and A which are 0.8 m apart. A particle P of mass 0.25 kg is projected with velocity $3 \mathrm{~m} \mathrm{~s}^{-1}$ horizontally from A in the direction away from O. The velocity of P is $v \mathrm{~m} \mathrm{~s}^{-1}$ when the displacement of P from O is $x \mathrm{~m}$. A force of magnitude $k v^{2} x^{-2} \mathrm{~N}$ opposes the motion of P.

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

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.