(a)[5]
Prove that $x$ and $t$ obey the differential equation $$\frac{dx}{dt}=\frac{-1}{2t(20x-3x^2)}.$
(b)[6]
Find a solution to the differential equation, giving $t$ in terms of $x$.
Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
A container shaped as a cuboid has a square base with side $x$ and height $(10-x)$. It is stated that $x$ changes with time, $t$, where $t>0$. The container's volume is falling at a rate inversely proportional to $t$.
When $t=\tfrac{1}{10}$, $x=\tfrac{1}{2}$ and the rate at which $x$ decreases is $\tfrac{20}{37}$.