Published September 2014.

Here you can find several example questions from STEP past papers for you to practice your skills on. There's questions covering formulating your own differential equation, as well as solving first order and second order problems: everything you need to get started. There's even hints to help you along if you need them - just click "show" to reveal them!

In this question, you may assume that $\ln (1+x) \approx x -\frac12 x^2$

when $\vert x \vert $ is small.

The height of the water in a tank at time $t$ is $h$.

The initial height of the water is $H$ and water flows into the tank at a constant rate.

The cross-sectional area of the tank is constant.

(i) Suppose that water leaks out at a rate proportional to the height of the water in the tank,

and that when the height reaches $\alpha^2 H$, where $\alpha$ is a constant greater than 1,

the height remains constant.

Show that $$ \frac {d h}{d t } = k( \alpha^2 H -h)\,$$

for some positive constant $k$. Deduce that the time $T$ taken for the water to reach height $\alpha H$ is given by

\[

kT = \ln \left(1+\frac1\alpha\right)\,,

\]

and that $kT\approx \alpha^{-1}$ for large values of $\alpha$.

(ii) Suppose that the rate at which water leaks out of the tank is proportional to $\sqrt h$ (instead of $h$), and that when the height reaches $\alpha^2 H$, where $\alpha$ is a constant greater than 1, the height remains constant.

Show that the time $T'$ taken for the water to reach height $\alpha H$ is

given by

$$cT'=2\sqrt H \left( 1 - \sqrt\alpha +\alpha \ln

\left(1+\frac1 {\sqrt\alpha} \right)\right)\,$$

for some positive constant $c$, and that $ cT'\approx \sqrt H$ for large values of $\alpha$.

(i) By writing $y=u{(1+x^2)\vphantom{\dot A}}^{\frac12}$, where $u$ is a function of $x$, find the solution of the equation $$\frac 1 y \frac{d y} {d x} = xy + \frac x {1+x^2}$$

for which $y=1$ when $x=0$.

(ii) Find the solution of the equation $$\frac 1 y \frac{d y} {d x} = x^2y + \frac {x^2 } {1+x^3}$$

for which $y=1$ when $x=0$.

(iii) Give, without proof, a conjecture for the solution of the equation $$\frac 1 y \frac{d y} {d x} = x^{n-1}y + \frac {x^{n-1} } {1+x^n}$$

for which $y=1$ when $x=0$, where $n$ is an integer greater than 1.

In order to find other solutions of this differential equation, now let $y=ue^x$, where $u$ is a function of $x$. By substituting this into $(*)$, show that $$(x-1) \frac{d^2 u}{d x^2} + (x-2) \frac{d u}{d x}=0\,.\tag{**}$$

By setting $ \dfrac {d u}{d x}= v$ in $(**)$ and solving the resulting first order differential equation for $v$, find $u$ in terms of $x$.

Hence show that $y=Ax + Be^x$ satisfies $(*)$, where $A$ and $B$ are any constants.