Prove that, if \[ y = A \cos(\log x) + B\sin(\log x), \] then \[ x^2 \frac{d^2y}{dx^2} + x\frac{dy}{dx} + y = 0. \]
Find the maxima and minima of the function \[ y = \sin x + \tfrac{1}{2}\sin 2x + \tfrac{1}{3}\sin 3x. \] Draw a graph of the function.
Integrate the functions \[ \frac{1}{x(x^2+a^2)}, \quad x^2\sin^2x, \quad e^x\cos 2x. \] Prove that \[ \int_0^1 x^2 \log x dx = -\tfrac{1}{9}. \]
Evaluate \[ \int \frac{x dx}{(x^2-a^2)^2+b^2x^2}, \quad a>0, b>0, \] distinguishing between the cases in which \(b<2a\) and \(b>2a\).
Sketch the curve defined by the equations \[ x=a\cos^3\theta, \quad y=a\sin^3\theta, \] and show that its total length is \(6a\).
Prove that, if \(\alpha\) is a constant, the function \[ y = A \cos\alpha x + B \sin\alpha x + \frac{1}{\alpha}\int_0^x f(\xi)\sin\alpha(x-\xi)d\xi \] satisfies the equation \[ \frac{d^2y}{dx^2} + \alpha^2y = f(x). \]
Three light strings are attached at points \(A\), \(B\), \(C\) to a circular hoop which is in a vertical plane where \(A\), \(B\), \(C\) are the vertices of an equilateral triangle, and the other ends of the strings support a particle of weight \(W\) at the centre of the hoop. If for any position of the hoop the tensions of the three strings are \(P, Q, R\), prove that \[ P^2+Q^2+R^2-QR-RP-PQ=W^2. \]
If known weights are attached to points on a light string, the ends of which are fixed, and if the directions of two portions of the string are known, shew how to find the direction of each portion of the string. \par Weights \(W, w, W\) are attached to points \(B, C, D\) respectively on a light string \(AE\), where \(B, C, D\) divide the string into four equal lengths. If the string can hang in the form of four sides of a regular octagon with the ends \(A\) and \(E\) attached to points on the same level, prove that \(W=(\sqrt{2}+1)w\).
Prove that a system of forces in a plane can be replaced by two forces in the plane, one acting along a given line and one acting through a given point. \par A rigid roof truss \(ABC\) is in the form of an isosceles triangle right-angled at \(B\) and rests on two walls at \(A\) and \(C\). It carries a weight \(W\) symmetrically distributed on the two sides, and due to wind pressure there is a force \(w\) uniformly distributed along \(BC\) perpendicular to \(BC\). If the reaction at \(C\) is vertical, find the horizontal and vertical components of the reaction at \(A\).
Establish the principle of virtual work for a lamina under the action of forces in its plane. \par Three similar uniform rods \(AB, BC, CD\) each of length \(a\) are smoothly jointed together at \(B\) and \(C\) and the ends \(A\) and \(D\) are smoothly jointed at points on the same level at a distance \(2a\) apart. A weight \(W\) hangs from a point of trisection of \(BC\) and the system is kept in equilibrium with \(BC\) horizontal by a force \(P\) acting along \(BC\). Prove that \(3\sqrt{3}P=W\), independently of the weight of the rods.