Practice section 3 of 6

[Radioactive Decay Modelling]

The rate of decrease of Iodine-131 is proportional to its amount x. Half the amount decays in 8.02 days. If the initial amount is x0, express x in terms of t and x0.

[Motion of Particles with Resistance]

Modified from Fundamentals of Differential Equations and Boundary Value Problems, sixth edition, p110. An object starts with velocity v0 and falls under gravity $g=9.81$ metres per second squared. Air resistance gives deceleration proportional to velocity. Given $dv/dt=g-kv$, $k>0$, express v in terms of t.

Show the general solution of $y\,dy/dx=2x-1$, $y\ge0$, is $y^2=2x^2-2x+C$. On one diagram sketch the members with $C=-1,0,1$.

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.