Practice section 2 of 6

Substitute $y=z-x$ into $(1/y)(dy/dx)-1=(x-2)/y$ to obtain an equation relating z and x. Solve it and hence give the general solution for y in terms of x.

Using $y=vx$, solve $x\,dy/dx=y+x^2$, where $y=0$ at $x=-1$, expressing y in terms of x.

Show $dy/dx=\sin^2(2x+y)-3$ reduces under $u=2x+y$ to $du/dx=-\cos^2u$. Hence find the solution through the origin as $y=f(x)$.

(a) The rate of increase of an insect population is proportional to the number n present at time t. Write the differential equation for n.

(b) A bird population has births per unit time proportional to population x and deaths per unit time proportional to its square. Form the equation for x.

[Newton’s Law of Cooling]

The rate of temperature loss is proportional to the difference between a body and its surroundings. Express this as a differential equation, using t for time and theta for body temperature.

A rod is immersed in water held at 15°C. It cools from 60°C to 40°C in 3 minutes. Find (i) its temperature at time t, and (ii) its temperature immediately after 9 minutes.

[Diseases modelling]

A community has 1000 people. The infected population x at t days satisfies $dx/dt=kx(1000-x)$, with $k>0$. Initially one person is infected; after five days 12% are infected. Find when half the population has contracted the disease and state an assumption of the model.