- Two compounds, \(X\) and \(Y\), are involved in a chemical reaction. The amounts in grams of these compounds, \(t\) minutes after the reaction starts, are \(x\) and \(y\) respectively and are modelled by the differential equations
$$\begin{aligned}
& \frac { \mathrm { d } x } { \mathrm {~d} t } = - 5 x + 10 y - 30
& \frac { \mathrm {~d} y } { \mathrm {~d} t } = - 2 x + 3 y - 4
\end{aligned}$$
- Show that
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 50$$
- Find, according to the model, a general solution for the amount in grams of compound \(X\) present at time \(t\) minutes.
- Find, according to the model, a general solution for the amount in grams of compound \(Y\) present at time \(t\) minutes.
Given that \(x = 2\) and \(y = 5\) when \(t = 0\)
- find
- the particular solution for \(x\),
- the particular solution for \(y\).
A scientist thinks that the chemical reaction will have stopped after 8 minutes.
- Explain whether this is supported by the model.