Differential equations

(i) Find the general solution of the differential equation $\left(\dfrac{x^2+1}{x}\right)\dfrac{dy}{dx}=2$.

Find the general solution of the differential equation $\dfrac{d^2y}{dx^2}=\sec x\tan x$.

Use the substitution $u=xy$, where $u$ is a function of $x$, to reduce the differential equation $x\dfrac{dy}{dx}+(1-x^3)y-x^2=0$ to $\dfrac{du}{dx}=(u+1)x^2$.

It is known that a particular type of bacteria grows very well under certain controlled conditions in a specially prepared Petri dish. The researcher believes that the growth rate of such bacteria can be modelled by $t^2\dfrac{dx}{dt}-2xt=-x^2$, where $x$ milligrams is the amount of bacteria grown in the dish after $t$ hours.

A spherical tank with negligible thickness and internal radius $a$ cm contains water. At time $t$ s, the water surface is at a height $x$ cm above the lowest point of the tank and the volume of water in the tank, $V$ cm³, is given by $V=\dfrac13\pi x^2(3a-x)$. Water flows from the tank, through an outlet at its lowest point, at a rate $\pi k\sqrt x$ cm³ s⁻¹,…

A metal ball is released from the surface of the liquid in a tall cylinder. The ball falls vertically through the liquid and the distance, $x$ cm, that the ball has fallen in time $t$ seconds is measured. The speed of the ball at time $t$ seconds is $v$ cm s⁻¹. The ball is released in a manner such that $x=0$ and $v=0$ when $t=0$.

A certain radioactive substance is known to decay at a rate proportional to the mass present. A block of this substance has an initial mass of 200 grams. After 40 hours, its mass is reduced to 180 grams. Given that $x$ grams is the mass of the substance present at time $t$ hours, write down a differential equation relating $x$ and $t$.

Alice bakes a large pie and removes it from the oven at 1 pm. The temperature reading of the pie using a food thermometer is 200 °C at that instant. She left the pie on the kitchen countertop to cool down. The temperature of the kitchen is kept at a constant 32 °C. The temperature of the pie $t$ minutes after it is removed from the oven is $\theta$ °C. This…

Glucose in the blood stream is reduced at a rate proportional to the amount of glucose present in the blood stream. Let $G$ milligrams (mg) be the amount of glucose in 1 decilitre (dL) of blood stream at time $t$ minutes and let $\lambda$ denote the positive constant of proportionality.

Jesse leans a 6 metre rod XY in the position as shown below. At time $t$ seconds, the two ends X and Y are $x$ and $y$ metres from O respectively.

Since the 1990s, a group of scientists has started conservation work to prevent the extinction of a rare species of flying fox in the wild Western Australia forests. The number of such species, in thousands, observed in the forests at time $t$ years after the start of the conservation is denoted by $N$. It is known that the death rate of the flying foxes is…

Botanists are studying how excessive logging is affecting the population of forest trees in a country in South-East Asia. The population of trees in a forest after $t$ years is denoted by $x$, in thousands. One simple model proposed is that the population increases at a rate proportional to its population size. It is known that trees are chopped off at a…

In a tank at an aquarium, fish pollute the water at a constant rate. A filter is installed into a new clean tank to remove the pollutant. The rate at which it removes the pollutant from the tank is proportional to the square root of the amount of pollutant present in the tank at that instant. Immediately after being cleaned of all pollutant, the amount of…

(a) By means of the substitution $z=xy$, show that the differential equation $2x\dfrac{dy}{dx}=\dfrac1x-y$ can be reduced to the form $\dfrac2{1+z}\dfrac{dz}{dx}=\dfrac1x$.

A scientist and his student are investigating the growth of bacteria in a petri dish. At time $t$ minutes after the bacteria is first introduced into the dish, the number of bacteria in the dish is denoted by $x$, in hundreds. Initially, 100 bacteria were introduced into the dish. In 10 minutes, the number of bacteria became 400.

On a mangrove swamp, scientists are investigating the population of mudskippers and crabs. Initially, there are 800 mudskippers and 800 crabs on the mangrove swamp. At time $t$ years, the number of mudskippers and crabs are $M,C$ respectively.

The cylindrical tank in a research laboratory has a cross-sectional area of 4 m². To cool the tank, water is pumped in and out of the tank simultaneously. The volume and height of the water in the tank at any time $t$ minutes is given by $V$ (m³) and $h$ (metres) respectively. Clean water is pumped into the tank at a rate that is proportional to $h^2$ and…

A flu virus is spreading in a community which has a fixed population of $N$ people. Scientists discover that at time $t$ weeks from the beginning, the rate of change of the number of people who are infected is proportional to the product of the number of people infected, $I$, and the number of people who are not infected.