Mathematics 9709 · AS & A Level · Differential equations
Differential equations — practice question
A certain substance is produced during a chemical reaction. Let the mass of substance produced after $t$ seconds from the start of the reaction be $x$ grams. At every instant, the rate at which the substance is formed is proportional to $(20 - x)$. When $t = 0$, $x = 0$ and $\frac{dx}{dt} = 1$.
(i)[2]
Show that $x$ and $t$ satisfy the differential equation $\frac{dx}{dt} = 0.05(20 - x)$.
(ii)[5]
Find, in any form, the solution of this differential equation.
(iii)[2]
Find $x$ when $t = 10$, with your answer correct to 1 decimal place.
(iv)[1]
State how the value of $x$ behaves as $t$ becomes very large.
Worked solution & mark scheme
This 10-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State or imply $\frac{dx}{dt}=k(20-x)$” …