If \(y = e^{ax^2}\) and \(u = \frac{d^n y}{dx^n}\), prove that \[ \frac{d^2u}{dx^2} - 2ax \frac{du}{dx} - (2n+2)au = 0. \] If \(u = e^{ax^2}v\), find a differential equation satisfied by \(v\). Shew that \(v\) is a polynomial of degree \(n\) in \(x\), and find the coefficient of \(x^{n-2}\).
Prove either of the two following theorems and deduce the other:
A motor car of mass 1 ton exerts a constant force of 100 lb. weight and has a maximum speed of 50 miles per hour on the level. Assuming that the frictional resistances are proportional to the square of the velocity, find the distance required for the car to accelerate from 10 to 20 miles per hour up an incline of 1 in 100.
Perform the following integrations: \[ \int \frac{e^{\sin^{-1} x}}{\sqrt{1-x^2}} dx, \quad \int \sqrt{\frac{e^x+a}{e^x-a}} dx, \quad \int \cosh mx \sin nx dx. \]
Sketch the curve \[ y^2 = \frac{2x-1}{x^2-1}. \] Shew that \(x+y=1\) is an inflexional tangent. Are there any others?
Prove that the mid-points of the sides of a triangle inscribed in a rectangular hyperbola \(H\) lie on a circle through the centre of \(H\).
A particle is free to move on a smooth vertical circle of radius \(a\). It is projected from the lowest point with velocity just sufficient to carry it to the highest point. Shew that, after a time \[ \sqrt{\frac{a}{g}} \log_e (\sqrt{5} + \sqrt{6}), \] the reaction between the particle and the wire is zero.
A cylinder A rolls without slipping on the outside of a fixed horizontal cylinder B, the generators remaining parallel. A is slightly disturbed from equilibrium in a position in which the common tangent plane is horizontal. Shew that the equilibrium is stable if \[ 1/h > 1/r_1 \pm 1/r_2, \] where \(h\) is the height of the centre of gravity of the body above the point of support, and \(r_1, r_2\) are the radii of curvature of the body and the fixed surface. What is the significance of the choice of sign? A uniform elliptic cylinder of semi-axes \(a\) and \(b\) (\(a > b\)) is placed on the top of a fixed rough circular cylinder of radius \(r\), the generators being parallel and horizontal. Shew that there are positions of equilibrium when the major axis of the cross-section is either horizontal or vertical, but that the second of these is always unstable. Shew also that the position in which the major axis is horizontal is stable only if \[ r > a^2b/(a^2 - b^2). \]
Prove that the evolute of the logarithmic spiral \(r=ae^{\alpha\theta}\) is an equal spiral.
Shew that chords of a conic \(S\) which subtend a right angle at a given point \(O\) of \(S\) pass through a fixed point \(F\) lying on the normal at \(O\). If \(S\) is a parabola, shew that as \(O\) describes \(S\), the point \(F\) describes an equal parabola.