4.10d Second order homogeneous: auxiliary equation method

1. The concentration of a drug in the bloodstream of a patient, \(t\) hours after the drug has been administered, where \(t \leqslant 6\), is modelled by the differential equation

$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } C } { \mathrm {~d} t ^ { 2 } } - 5 t \frac { \mathrm {~d} C } { \mathrm {~d} t } + 8 C = t ^ { 3 }$$ where \(C\) is measured in micrograms per litre.

1. Show that the transformation \(t = \mathrm { e } ^ { x }\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} C } { \mathrm {~d} x } + 8 C = \mathrm { e } ^ { 3 x }$$
2. Hence find the general solution for the concentration \(C\) at time \(t\) hours. Given that when \(t = 6 , C = 0\) and \(\frac { \mathrm { d } C } { \mathrm {~d} t } = - 36\)
3. find the maximum concentration of the drug in the bloodstream of the patient.

1. A vibrating spring, fixed at one end, has an external force acting on it such that the centre of the spring moves in a straight line. At time \(t\) seconds, \(t \geqslant 0\), the displacement of the centre \(C\) of the spring from a fixed point \(O\) is \(x\) micrometres.

The displacement of \(C\) from \(O\) is modelled by the differential equation $$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 2 + t ^ { 2 } \right) x = t ^ { 4 }$$

1. Show that the transformation \(x = t v\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} t ^ { 2 } } + v = t$$
2. Hence find the general equation for the displacement of \(C\) from \(O\) at time \(t\) seconds.
3. 1. State what happens to the displacement of \(C\) from \(O\) as \(t\) becomes large.
   2. Comment on the model with reference to this long term behaviour.

8. $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 2 & 5 \\ 0 & 3 & p \\ - 6 & 6 & - 4 \end{array} \right) \quad \text { where } p \text { is a constant }$$ Given that \(\left( \begin{array} { r } 2 \\ 1 \\ - 2 \end{array} \right)\) is an eigenvector for \(\mathbf { A }\)

1. 1. determine the eigenvalue corresponding to this eigenvector
   2. hence show that \(p = 2\)
   3. determine the remaining eigenvalues and corresponding eigenvectors of \(\mathbf { A }\)
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } = \mathbf { P D P } ^ { - 1 }\)
3. 1. Solve the differential equation \(\dot { u } = k u\), where \(k\) is a constant. With respect to a fixed origin \(O\), the velocity of a particle moving through space is modelled by $$\left( \begin{array} { c } \dot { x } \\ \dot { y } \\ \dot { z } \end{array} \right) = \mathbf { A } \left( \begin{array} { l } x \\ y \\ z \end{array} \right)$$ By considering \(\left( \begin{array} { c } u \\ v \\ w \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } x \\ y \\ z \end{array} \right)\) so that \(\left( \begin{array} { c } \dot { u } \\ \dot { v } \\ \dot { w } \end{array} \right) = \mathbf { P } ^ { - 1 } \left( \begin{array} { c } \dot { x } \\ \dot { y } \\ \dot { z } \end{array} \right)\)
   2. determine a general solution for the displacement of the particle.

1. Determine a closed form for the recurrence relation

$$\begin{aligned} & u _ { 0 } = 1 \quad u _ { 1 } = 4 \\ & u _ { n + 2 } = 2 u _ { n + 1 } - \frac { 4 } { 3 } u _ { n } + n \quad n \geqslant 0 \end{aligned}$$

1. Determine a closed form for the recurrence system

$$\begin{gathered} u _ { 1 } = 4 \quad u _ { 2 } = 6 \\ u _ { n + 2 } = 6 u _ { n + 1 } - 9 u _ { n } \quad ( n = 1,2,3 , \ldots ) \end{gathered}$$

1. A staircase has \(n\) steps. A tourist moves from the bottom (step zero) to the top (step \(n\) ). At each move up the staircase she can go up either one step or two steps, and her overall climb up the staircase is a combination of such moves.

If \(u _ { n }\) is the number of ways that the tourist can climb up a staircase with \(n\) steps,

1. explain why \(u _ { n }\) satisfies the recurrence relation $$u _ { n } = u _ { n - 1 } + u _ { n - 2 } , \text { with } u _ { 1 } = 1 \text { and } u _ { 2 } = 2$$
2. Find the number of ways in which she can climb up a staircase when there are eight steps. A staircase at a certain tourist attraction has 400 steps.
3. Show that the number of ways in which she could climb up to the top of this staircase is given by $$\frac { 1 } { \sqrt { 5 } } \left[ \left( \frac { 1 + \sqrt { 5 } } { 2 } \right) ^ { 401 } - \left( \frac { 1 - \sqrt { 5 } } { 2 } \right) ^ { 401 } \right]$$

6 One end of a light inextensible string is attached to a small mass. The other end is attached to a fixed point \(O\). Initially the mass hangs at rest vertically below \(O\). The mass is then pulled to one side with the string taut and released from rest. \(\theta\) is the angle, in radians, that the string makes with the vertical through \(O\) at time \(t\) seconds and \(\theta\) may be assumed to be small. The subsequent motion of the mass can be modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - 4 \theta$$

1. Write down the general solution to this differential equation.
2. Initially the pendulum is released from rest at an angle of \(\theta _ { 0 }\). Find the particular solution to the equation in this case.
3. State any limitations on the model.

3 A particle of mass 2 kg moves along the \(x\)-axis. At time \(t\) seconds the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\). The particle is subject to two forces.

• One acts in the positive \(x\)-direction with magnitude \(\frac { 1 } { 2 } t \mathrm {~N}\).
• One acts in the negative \(x\)-direction with magnitude \(v \mathrm {~N}\).
  1. Show that the motion of the particle can be modelled by the differential equation

$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$ The particle is at rest when \(t = 0\).

Find \(v\) in terms of \(t\).

Find the velocity of the particle when \(t = 2\). When \(t = 2\) the force acting in the positive \(x\)-direction is replaced by a constant force of magnitude \(\frac { 1 } { 2 } \mathrm {~N}\) in the same direction.

Refine the differential equation given in part (a) to model the motion for \(t \geqslant 2\).

Use the refined model from part (d) to find an exact expression for \(v\) in terms of \(t\) for \(t \geqslant 2\).

8 For the differential equation \(t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 6 - 4 t ^ { 2 } \right) x = 0\), use the substitution \(x = t ^ { 2 } u\) to find a differential equation involving \(t\) and \(u\) only. Hence solve the above differential equation, given that \(x = \mathrm { e } - 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 4 \mathrm { e }\) when \(t = 1\).

10

1. Given that \(y = k x \cos x\) is a particular integral for the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y = 4 \sin x$$ determine the value of \(k\) and find the general solution of this differential equation.
2. The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 5 x - 19$$
   1. Given that \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 1\), find the value of \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\) when \(x = 1\).
   2. Deduce the Taylor series expansion for \(y\) in ascending powers of \(( x - 1 )\), up to and including the term in \(( x - 1 ) ^ { 3 }\), and use this series to find an approximation correct to 3 decimal places for the value of \(y\) when \(x = 1.1\).

5 Find the general solution of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 24 \mathrm { e } ^ { 2 x }\).

Find the general solution of the differential equation $$\frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = e^{-\alpha t},$$ where \(\alpha\) is a constant and \(\alpha \neq 2\). [7] Show that if \(\alpha < 2\) then, whatever the initial conditions, \(ye^{\alpha t} \to \frac{1}{(2-\alpha)^2}\) as \(t \to \infty\). [2]

Solve the differential equation $$\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = 24e^{2x},$$ given that \(y = 1\) and \(\frac{dy}{dx} = 9\) when \(x = 0\). [7]

Find the general solution of the differential equation $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 4\frac{\mathrm{d}x}{\mathrm{d}t} + 4x = 7 - 2t^2.$$ [6]

Find the particular solution of the differential equation $$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 3y = 27x^2,$$ given that, when \(x = 0\), \(y = 2\) and \(\frac{dy}{dx} = -8\). [10]

Find the particular solution of the differential equation $$3\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = x^2,$$ given that, when \(x = 0\), \(y = \frac{dy}{dx} = 0\). [10]

$$\frac{d^2 y}{dt^2} - 6\frac{dy}{dt} + 9y = 4e^{3t}, \quad t \geq 0.$$

1. Show that \(Kte^{3t}\) is a particular integral of the differential equation, where \(K\) is a constant to be found. [4]
2. Find the general solution of the differential equation. [3] Given that a particular solution satisfies \(y = 3\) and \(\frac{dy}{dt} = 1\) when \(t = 0\),
3. find this solution. [4] Another particular solution which satisfies \(y = 1\) and \(\frac{dy}{dt} = 0\) when \(t = 0\), has equation $$y = (1 - 3t + 2t^2)e^{3t}.$$
4. For this particular solution draw a sketch graph of \(y\) against \(t\), showing where the graph crosses the \(t\)-axis. Determine also the coordinates of the minimum of the point on the sketch graph. [5]

$$\frac{d^2 y}{dx^2} + 4\frac{dy}{dx} + 5y = 65 \sin 2x, \quad x > 0.$$

1. Find the general solution of the differential equation. [9]
2. Show that for large values of \(x\) this general solution may be approximated by a sine function and find this sine function. [2]