Practice section 4 of 6

Use $u=y/x$ to show that $\dfrac{dy}{dx}=\dfrac{y^2+xy+x^2}{x^2}$, $x>0$, reduces to $\dfrac1{u^2+1}\dfrac{du}{dx}=\dfrac1x$. Hence find $y$ in terms of $x$.

Find the particular solution of $(e^{2y}-y)\cos x\,\dfrac{dy}{dx}=e^y\sin2x$, given $y=0$ when $x=0$.

In a favourable environment, bacteria increase at a rate proportional to the number present. There are $1\,000\,000$ at a certain time and $2\,000\,000$ one hour later. Find the number present four hours later.

When a metal rod is heated, its length $L$ increases at a rate proportional to $L$.

(i) Write a differential equation for $dL/dt$ in terms of $L$.

(ii) Show that its general solution is $L=Ae^{kt}$.

(iii) The initial length is $50$ cm. Sketch $L$ against $t$.

(iv) Explain why this differential equation does not appropriately describe the expansion of the heated rod.

A bottle of liquid is removed from a refrigerator into a room at constant $20^\circ$C. Its temperature $\theta$ degrees Celsius after $t$ minutes increases at a rate proportional to $(20-\theta)$. Initially $\theta=10$ and $d\theta/dt=1$ degree Celsius per minute. Set up and solve a differential equation to show that $\theta=20-10e^{-t/10}$. Find the time to reach $15^\circ$C, state the large-$t$ behaviour, and sketch $\theta$ against $t$.

During rainfall, a reservoir contains $x$ million cubic metres at time $t$ hours. Inflow is constant at $k$ million cubic metres per hour, and outflow is proportional to $x^2$.

(i) Given that the water amount stabilises at $0.5$ million cubic metres, show that $dx/dt=k(1-4x^2)$.

(ii) Find $x$ in terms of $k,t$ if initially there is $1$ million cubic metre.