Complex roots giving oscillatory solution

A question is this type if and only if the auxiliary equation of the derived second-order ODE yields complex roots, requiring the general solution to be expressed in terms of exponentials multiplied by sine and cosine.

2 questions · Challenging +1.2

4.10e Second order non-homogeneous: complementary + particular integral4.10h Coupled systems: simultaneous first order DEs
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Edexcel CP1 2020 June Q5
17 marks Challenging +1.2
  1. 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
$$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = - 5 x + 10 y - 30 \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = - 2 x + 3 y - 4 \end{aligned}$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 50$$
  2. Find, according to the model, a general solution for the amount in grams of compound \(X\) present at time \(t\) minutes.
  3. 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 \(t = 0\)
  4. find
    1. the particular solution for \(x\),
    2. the particular solution for \(y\). A scientist thinks that the chemical reaction will have stopped after 8 minutes.
  5. Explain whether this is supported by the model.
Edexcel CP2 2023 June Q9
14 marks Challenging +1.2
  1. A patient is treated by administering an antibiotic intravenously at a constant rate for some time.
Initially there is none of the antibiotic in the patient.
At time \(t\) minutes after treatment began
  • the concentration of the antibiotic in the blood of the patient is \(x \mathrm { mg } / \mathrm { ml }\)
  • the concentration of the antibiotic in the tissue of the patient is \(y \mathrm { mg } / \mathrm { ml }\)
The concentration of antibiotic in the patient is modelled by the equations $$\begin{aligned} & \frac { \mathrm { d } x } { \mathrm {~d} t } = 0.025 y - 0.045 x + 2 \\ & \frac { \mathrm {~d} y } { \mathrm {~d} t } = 0.032 x - 0.025 y \end{aligned}$$
  1. Show that $$40000 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2800 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 13 y = 2560$$
  2. Determine, according to the model, a general solution for the concentration of the antibiotic in the patient's tissue at time \(t\) minutes after treatment began.
  3. Hence determine a particular solution for the concentration of the antibiotic in the tissue at time \(t\) minutes after treatment began. To be effective for the patient the concentration of antibiotic in the tissue must eventually reach a level between \(185 \mathrm { mg } / \mathrm { ml }\) and \(200 \mathrm { mg } / \mathrm { ml }\).
  4. Determine whether the rate of administration of the antibiotic is effective for the patient, giving a reason for your answer.