If \(y = \sin(x \sin^{-1} x)\), prove \[(1-x^2) y'' - xy' + x^2 y = 0,\] where \(y'\) and \(y''\) represent the first and second derivatives of \(y\). Prove that the Maclaurin series of \(y\) is \[x\left[x + \frac{(1^2-x^2)x^3}{3!} + \frac{(3^2-x^2)(1^2-x^2)x^5}{5!} + \ldots\right].\]
(i) Prove \[\int_0^a \frac{x^2 dx}{x^2 + (x-a)^2} = \int_0^a \frac{(x-a)^2 dx}{x^2 + (x-a)^2} = \frac{1}{2}a.\] (ii) Evaluate \[\int x^3 \tan^{-1} x dx.\] (iii) Given \(a > b > 0\), evaluate \[\int_0^\pi \frac{\cos x dx}{a^2 + b^2 - 2ab \cos x}.\]
Two variables \(x\) and \(y\) are to be determined as functions of time \(t\). It is found that the rate of change of \(x\) is equal to the sum of \(k_1\) times the instantaneous value of \(x\) and \((-k_2)\) times the instantaneous value of \(y\). The rate of change of \(y\) is similarly equal to \((-k_3)\) times the value of \(x\) plus \(k_4\) times the value of \(y\). Here \(k_1\), \(k_2\), \(k_3\) and \(k_4\) are positive constants. Obtain a second-order differential equation for \(x(t)\) and show that if \(k_1 k_4 > k_2 k_3\) the solution is of the form \[x = A e^{\alpha t} + B e^{\beta t},\] where \(A\) and \(B\) are arbitrary constants and \(\alpha\) and \(\beta\) are positive.
(i) Given \(\arg(z + a) = \frac{1}{4}\pi\) and \(\arg(z - a) = \frac{3}{4}\pi\), where \(a\) is a given real positive number, find the complex number \(z\). [\(\arg z = 0\) means that \(z\) is a real positive multiple of \(\cos \theta + i \sin \theta\).] (ii) Given \[|z + c| + |z - c| < 2d,\] where \(c\) is complex and \(d > |c|\), and also \[\pi < \arg z < 2\pi,\] describe geometrically the region of the complex plane in which \(z\) must lie.
Show that \(x \tan x = 1\) has an infinite number of real roots, and that if \(n\) is a large integer there is a root near \(n\pi\). Show that a better approximation is \(n\pi + (1/n\pi)\), and find a better one still.
Show that \(y = \sin x \tan x - 2 \log \sec x\) increases steadily as \(x\) increases from \(0\) to \(\frac{1}{2}\pi\). Show also that \(y\) has no inflexion in this range. Sketch the curve \(y(x)\) in \[0 \leq x < \frac{1}{2}\pi.\]
The region \(A_n\) of the \((x,y)\)-plane is bounded by the portions of the curves \(y = 0\) and \(y = \sin^n x\) given by \(0 \leq x \leq \pi\), where \(n\) is a positive integer. Show that, if \(n > 2\), the area \(a_n\) of \(A_n\) satisfies \[na_n = (n-1)a_{n-2},\] and hence find \(a_n\) for all \(n \geq 1\). Deduce that \(a_{2n}/a_{2n+1} \to 1\) as \(n \to \infty\), and hence show that \[\frac{2}{2n+1} \frac{2^{4n}(n!)^4}{[(2n)!]^2} \to \pi \quad \text{as} \quad n \to \infty.\]
By use of the identity \[(1+y)(1-y+y^2-\ldots+(-y)^n) \equiv 1-(-y)^{n+1},\] or otherwise, prove that, for any \(n > 0\), the value of \(\tan^{-1} x\) in the range \((-\frac{1}{2}\pi, \frac{1}{2}\pi)\) lies between \(s_n\) and \(s_{n+1}\), where \[s_n = \sum_{r=1}^n \frac{(-1)^{r+1} x^{2r-1}}{2r-1}.\] Show that \[\sum_{n=1}^\infty \frac{(-1)^{n+1}}{(2n-1) \cdot 3^n}\] converges to \(\pi/(6\sqrt{3})\).
Define the curvature \(\kappa\) at a point of a curve having a smoothly-turning tangent. Show that, if the rectangular Cartesian coordinates \((x,y)\) of the general point of the curve are given as functions of a parameter \(\theta\), then \[\kappa = \frac{x'y'' - y'x''}{[(x')^2 + (y')^2]^{3/2}},\] where dashes denote differentiations with respect to \(\theta\). Find the curvature at the general point of the curve \begin{align} x &= a \cos \theta + a\theta \sin \theta,\\ y &= a \sin \theta - a\theta \cos \theta. \end{align} Verify your result by showing that the normal at any point of the curve touches a circle \(x^2 + y^2 = a^2\). Deduce a mechanical method of drawing the curve, and sketch that part corresponding to values of \(\theta\) in the range \([0, 2\pi]\).
A triangular lamina is given, and instruments capable of measuring lengths and angles to within known small margins of error (absolute, not percentage). If you were asked to calculate the radius of the circumcircle of the triangle from measurements of one side and one angle of the triangle, which would you choose in order to obtain the most reliable answer? Why? A number of triangular laminas of widely varying shapes are given, all of whose perimeters lie between 15 and 16 inches. The instruments provided can measure lengths to within 0.1 inch and angles to within 0.001 radian. For which shape would you expect the percentage error in your estimate (by the above method) of the radius of the circumcircle to be greatest? Why?