Discussion questions · Section 1 of 2

Find the general solution of the differential equation $\dfrac{dy}{dx}=10x-3x^2$, expressing $y$ in terms of $x$. Given that $y=100$ when $x=0$, sketch this curve.

Show that $y=ux^2$ reduces $x^2\dfrac{dy}{dx}-2xy+y^2=0$, where $x>0$, to $\dfrac{du}{dx}+u^2=0$. Hence show that the general solution may be expressed as $y=x^2/(C+x)$, with arbitrary constant $C$.

The variables satisfy $xe^y\dfrac{dy}{dx}+e^y=2x$.

(i) Use $z=xe^y$ to obtain an equation relating $z$ and $x$.

(ii) Hence, or otherwise, show that the general solution is $y=\ln(x+k/x)$, where $k$ is arbitrary.

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.