Standard +0.3 This is a standard separable differential equation requiring partial fractions (given in part a), separation of variables, integration, and applying initial conditions. While it involves multiple steps, each technique is routine for Further Maths students. The partial fractions are straightforward, the integration is standard, and parts (c) and (d) are simple substitution/limit evaluation. Slightly easier than average due to the scaffolding provided.
14. a. Express \(\frac { 1 } { ( 3 - x ) ( 1 - x ) }\) in partial fractions.
(2)
A scientist is studying the mass of a substance in a laboratory.
The mass, \(x\) grams, of a substance at time \(t\) seconds after a chemical reaction starts is modelled by the differential equation
$$2 \frac { d x } { d t } = ( 3 - x ) ( 1 - x ) \quad t \geq 0,0 \leq x < 1$$
Given that when \(t = 0 , x = 0\)
b. solve the differential equation and show that the solution can be written as
$$x = \frac { 3 \left( e ^ { t } - 1 \right) } { 3 e ^ { t } - 1 }$$
c. Find the mass, \(x\) grams, which has formed 2 seconds after the start of the reaction. Give your answer correct to 3 significant figures.
d. Find the limiting value of \(x\) as \(t\) increases.
14. a. Express $\frac { 1 } { ( 3 - x ) ( 1 - x ) }$ in partial fractions.\\
(2)
A scientist is studying the mass of a substance in a laboratory.\\
The mass, $x$ grams, of a substance at time $t$ seconds after a chemical reaction starts is modelled by the differential equation
$$2 \frac { d x } { d t } = ( 3 - x ) ( 1 - x ) \quad t \geq 0,0 \leq x < 1$$
Given that when $t = 0 , x = 0$\\
b. solve the differential equation and show that the solution can be written as
$$x = \frac { 3 \left( e ^ { t } - 1 \right) } { 3 e ^ { t } - 1 }$$
c. Find the mass, $x$ grams, which has formed 2 seconds after the start of the reaction. Give your answer correct to 3 significant figures.\\
d. Find the limiting value of $x$ as $t$ increases.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q14 [9]}}