For $x>0$, a curve satisfies

and passes through $(1,2)$.

The temperature $T$, in degrees Celsius, of a heated object is modelled by

where $t$ is measured in minutes. Initially $T=90$.

A radioactive substance has mass $M$ grams after $t$ years. Its decay is modelled by

After $200$ years, $65\%$ of the original mass remains.

A population $n$ is modelled by the logistic differential equation $\dfrac{\mathrm{d}n}{\mathrm{d}t}=\dfrac1{12}n(6-n)$, where $0<n<6$. Initially $n=2$ when $t=0$.

For $x>0$, a function $y$ satisfies $\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{y}{x}+\dfrac{x}{y}$ and $y(1)=2$.

The temperature $T$, in degrees Celsius, of a cup of tea after $t$ minutes is modelled by $\dfrac{\mathrm{d}T}{\mathrm{d}t}=-k(T-20)$, where $k>0$. Initially $T=80$, and after one minute $T=50$.

For $x>0$, the function $y$ satisfies

with $y(1)=1$.

The function $y$ satisfies the differential equation

with $y(0)=3$.

For $x>0$, a function $y$ satisfies

and $y(1)=0$.

For $x>0$, a curve satisfies

and passes through $(1,1)$.

A quantity $S$ satisfies the differential equation

where $t\ge 0$, and $S(0)=20$.

A curve satisfies the differential equation

with initial condition $y(0)=1$.

A function $y$ satisfies

For $x>0$, the function $y$ satisfies $x\dfrac{\mathrm{d}y}{\mathrm{d}x}+(x+1)y=x^2e^{-x}$, with $y(1)=0$.

A quantity $n$ is modelled by $\dfrac{\mathrm{d}n}{\mathrm{d}t}=\dfrac1{100}n(20-n)$, where $0<n<20$. Initially $n=5$ when $t=0$.

A curve satisfies

A population $P$ is modelled by the logistic differential equation

where $t$ is measured in days. Initially $P=50$, and after $10$ days $P=150$.

The differential equation

has initial condition $y(0)=1$.

A population $N$, where $0<N<10$, is modelled by

For $x>0$, a function $y$ satisfies

A function $y$ satisfies

Let $z=\frac1y$.

The function $y$ satisfies

A quantity $P$ satisfies

and $P=5$ when $t=0$.

For $x>0$, a curve satisfies

Use the substitution $y=vx$.

A function $y$ satisfies

A colony of bacteria is modelled by the logistic differential equation

where $P$ is measured in thousands and $t$ is measured in hours. Initially $P=1.2$ when $t=0$.

The temperature $T$, in degrees Celsius, of an industrial component after $t$ minutes is modelled by

A population $N$ is modelled by

where $t$ is measured in days. Initially $N=40$, and after $6$ days $N=160$.

For $x>0$, the function $y$ satisfies

The amount $M$, in milligrams, of a medicine in the bloodstream after $t$ hours is modelled by

For $x\ge 0$, a function $y$ satisfies

A model for the height $y$ metres of a fast-growing plant after $x$ weeks satisfies

The diagram shows four representative slope segments for this differential equation on the interval $0\le x\le 1$.

The number $N$ of cells, in thousands, in a culture is modelled by the logistic differential equation

where $t$ is measured in days. Initially $N=100$, and after $5$ days $N=250$.

A tank initially contains $50$ litres of solution with $5$ kg of dissolved salt. Brine enters at $3$ litres per minute with salt concentration $0.4$ kg per litre. The well-mixed solution leaves at $2$ litres per minute.

Let $S$ be the mass of salt, in kg, in the tank after $t$ minutes.

A metal probe has temperature $T$ degrees Celsius at time $t$ minutes. The surrounding air temperature is rising linearly and is given by

The probe is modelled by Newton's law of cooling,