Obtain, by the method of solution in series (using series of ascending powers of \(x\)), the complete primitive of the differential equation \[ (1-x^2)\frac{d^2y}{dx^2} + 2x\frac{dy}{dx} - 6y = 0. \] Taking \(x\) as real, examine carefully the range of validity of your solution.
A small magnet is placed at the centre of a spherical shell of magnetic material whose internal and external radii are \(a\) and \(b\). Find expressions for the magnetic potential in the three regions into which space is divided by the two spherical surfaces.
Prove that the intrinsic equation which represents the curve taken up by a uniform thin rod, when bent into a bow by means of a string attached to its ends, is \[ s = c \int_0^\psi \frac{d\psi}{\sqrt{\cos\psi-\cos\alpha}}, \] where \(c\) is a constant. \(s\), the distance along the curve, is measured from the middle point of the rod. \(\psi\) is the angle between the tangent at any point and the tangent at the middle of the rod. \(\pm\alpha\) are the values of \(\psi\) at the ends of the rod. Find the law according to which the diameter of a thin rod of circular section must taper towards its ends in order that it may form a circle when bent so that the ends come together, and are held in that position by a flexible joint.
A particle is acted on by a central force which varies inversely as the \(n\)th power of the distance. It is projected from a point at distance \(c\) from the centre in a direction making an angle \(\beta\) with the radius vector. The initial velocity is that which it would acquire after falling freely from rest at infinity to the point of projection. Show that the equation to the orbit is \[ \left(\frac{r}{c}\right)^{\frac{n-3}{2}} = \text{cosec } \beta \sin\left(\beta - \frac{n-3}{2}\theta\right) \] where \(\theta\) is measured from the radius vector of the point of projection.
Show how Lagrange's equations of motion may be used to determine the small oscillations of a dynamical system with a finite number of degrees of freedom. A uniform rod of length \(2a\) is hung horizontally by means of two vertical strings from two fixed points at distances \(l\) and \(l'\) above the ends of the rod. Show that the three periods of the normal oscillations are \[ 2\pi\sqrt{\left\{ g\left(\frac{1}{l}+\frac{1}{l'}\right) \pm g\sqrt{\left(\frac{1}{l}+\frac{1}{l'}\right)^2 + \frac{12}{a^2}\left(\frac{1}{l}-\frac{1}{l'}\right)^2} \right\}^{-1}} \text{ and } 2\pi\sqrt{\frac{1}{g}\left(\frac{1}{l}+\frac{1}{l'}\right)^{-1}}. \]
Straight ripples move along the surface of a liquid of infinite depth under the influence of gravity and capillarity. Find the wave velocity. Show that corresponding to any given wave speed there are two possible wave lengths provided that the speed is greater than a certain fixed value; but that there is only one wave length corresponding to a given frequency.
Assign two different possible meanings to the word ``random'' in the following question, and give the corresponding answers:— ``A chord is chosen at random in a circle. Find the probability that its length exceeds a side of the inscribed equilateral triangle.'' A stick of length \(l\) is dropped without rotation through a horizontal grating of parallel wires placed at a distance \(a\) apart. The direction in which the stick points is chosen entirely at random. Show that the chance that the stick will fall through the grating without touching any of the wires is \(1 - l/2a\) if \(a\) is greater than \(l\) and \(a/2l\) otherwise.