4.10d Second order homogeneous: auxiliary equation method

Solve the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} - 7\frac{\mathrm{d}y}{\mathrm{d}x} + 10y = 0,$$ where \(\frac{\mathrm{d}y}{\mathrm{d}x} = 1\) and \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} = 8\) when \(x = 0\). [10]

A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by \(Q\). The capacitor is placed in an electrical circuit. At any time \(t\) seconds, where \(t \geq 0\), \(Q\) can be modelled by the differential equation $$\frac{d^2Q}{dt^2} - 2\frac{dQ}{dt} - 15Q = 0.$$ Initially the charge is 100 units and it is given that \(Q\) tends to a finite limit as \(t\) tends to infinity.

1. Determine the charge on the capacitor when \(t = 0.5\). [6]
2. Determine the finite limit of \(Q\) as \(t\) tends to infinity. [1]

A swing door is a door to a room which is closed when in equilibrium but which can be pushed open from either side and which can swing both ways, into or out of the room, and through the equilibrium position. The door is sprung so that when displaced from the equilibrium position it will swing back towards it. The extent to which the door is open at any time, \(t\) seconds, is measured by the angle at the hinge, \(\theta\), which the plane of the door makes with the plane of the equilibrium position. See the diagram below. In an initial model of the motion of a certain swing door it is suggested that \(\theta\) satisfies the following differential equation. $$4\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + 25\theta = 0 \quad (*)$$

1. 1. Write down the general solution to (*). [2]
   2. With reference to the behaviour of your solution in part (a)(i) explain briefly why the model using (*) is unlikely to be realistic. [1]

In an improved model of the motion of the door an extra term is introduced to the differential equation so that it becomes $$4\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2} + \lambda\frac{\mathrm{d}\theta}{\mathrm{d}t} + 25\theta = 0 \quad (\dagger)$$ where \(\lambda\) is a positive constant.

1. In the case where \(\lambda = 16\) the door is held open at an angle of \(0.9\) radians and then released from rest at time \(t = 0\).
   1. Find, in a real form, the general solution of (\(\dagger\)). [3]
   2. Find the particular solution of (\(\dagger\)). [4]
   3. With reference to the behaviour of your solution found in part (b)(ii) explain briefly how the extra term in (\(\dagger\)) improves the model. [2]
2. Find the value of \(\lambda\) for which the door is critically damped. [2]

The differential equation \((\star)\) is $$\frac{\text{d}^2 u}{\text{d}x^2} + 4u = 8x + 1.$$

1. Find the general solution of \((\star)\). [5]
2. The differential equation \((\star \star)\) is $$x \frac{\text{d}^2 v}{\text{d}x^2} + 2 \frac{\text{d}v}{\text{d}x} + 4xv = 8x + 1.$$ By using the substitution \(u = xv\), show that \((\star)\) becomes \((\star \star)\) and deduce the general solution of \((\star \star)\). [4]

Find the general solution of the differential equation $$\frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + 6\frac{\mathrm{d}y}{\mathrm{d}x} + 9y = 72\mathrm{e}^{3x}.$$ [7]

11 Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right) .$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\). \end{document}