The coefficients \(a_1\) and \(a_2\) of the differential equation $$\frac{d^2y}{dx^2} + a_1 \frac{dy}{dx} + a_2y = 0$$ are real numbers. Write down the general real solution of the equation. It is given that every real solution of this equation is bounded for \(x \geq 0\) — that is, if \(f(x)\) is a real solution, there exists a constant \(M\) such that \(|f(x)| \leq M\) for all \(x \geq 0\). Show that \(a_1\) and \(a_2\) must be non-negative.
The sequence \(a_0, a_1, a_2, \ldots\) is defined by the recurrence relation $$a_0 = b,$$ $$a_{n+1} = \frac{1}{2}\left(\frac{c}{a_n} + a_n\right) \text{ for } n = 0, 1, 2, \ldots,$$ where \(b\) and \(c\) are positive numbers. Show that \(a_n\) tends to a limit as \(n \to \infty\), and identify the limit. [You may assume that a decreasing sequence of positive numbers tends to a limit.]
Spacecraft land on a spherical planet of centre \(O\). Each is able to transmit messages to, and receive messages from, any spacecraft on the half of the surface of the planet nearest to it. (i) It is known that spacecraft have landed at points \(A\) and \(B\) of the surface of the planet. Show that the probability that a spacecraft, landing at random on the planet, will be able to communicate directly with the spacecraft at \(A\) and \(B\) is $$\frac{\pi - \theta}{2\pi},$$ where \(\theta\) is the angle \(AOB\). (ii) What is the probability that three spacecraft, all landing at random on the planet, will be in direct contact with each other?
The function \(I(x)\) is defined for \(x > 0\) by $$I(x) = \int_1^x \frac{dt}{t}.$$ Show that \(I(xy) = I(x) + I(y)\). Show, by making the change of variables \(u = (1-\theta)t + \theta\), that if \(0 < \theta < 1\) and \(x > 1\), then $$(1-\theta)I(x) < I(\theta + (1-\theta)x).$$ Deduce that if \(0 < \theta < 1\) and \(0 < a \leq b\) then $$\theta I(a) + (1-\theta)I(b) \leq I(\theta a + (1-\theta)b).$$ What information does this inequality give about the shape of the graph of the function \(I(x)\)?
\(a_0, a_1, a_2, \ldots\) is a sequence of real numbers. Explain carefully what the following statements mean: \begin{align} (i) \quad & a_n \to a \text{ as } n \to \infty;\\ (ii) \quad & \sum_{n=0}^{\infty} a_n = b. \end{align} Show from first principles that if \(|x| < 1\) then \(x^n \to 0\) as \(n \to \infty\), and $$\sum_{n=0}^{\infty} x^n = (1-x)^{-1}.$$ [You may find the substitution \(|x| = 1/(1+y)\) helpful.]
The function \(f(x)\) is defined on the interval \(0 < x < 1\) as follows: (a) if \(x\) is rational, and \(x = p/q\) in lowest terms, then \(f(x) = 1/q\); (b) if \(x\) is irrational, \(f(x) = 0\). Show that if \(\epsilon\) is greater than \(0\), there are only finitely many points of the interval for which \(f(x) \geq \epsilon\). Deduce that \(f\) is continuous at the irrational points of the interval, and is discontinuous at the rational points.
A car has two gears, and its performance (after allowing for air resistance and friction) is such that in bottom gear the acceleration is \(2C(V-v)\), and in top gear it is \(C(V-\frac{1}{2}v)\). Here \(v\) is the speed of the car, \(C\) and \(V\) are constants. The car is started from rest and accelerated as quickly as possible, the gear change occupying negligible time. Show that a speed \(v = 4V/3\) is reached after time \(C^{-1}\log 4/3\), and calculate the distance travelled by then.
A plane lamina is acted on by forces having components \((X_r, Y_r)\) at points \((x_r, y_r)\) \((r = 1, 2, \ldots)\), in Cartesian coordinates. Writing \(z_r = x_r + iy_r\) and \(Z_r = X_r + iY_r\) (so that the points and forces may be represented in the complex plane), write down the complex number representing the resultant force, and show that the moment of the system about the point \((a, b)\) is $$-\mathcal{I}\sum_r[(z_r-c)\bar{Z_r}],$$ where \(c = a + ib\), the bar denotes the complex conjugate, and \(\mathcal{I}\) denotes the imaginary part. Using these formulae, or otherwise, show that if all the forces are turned through the same angle in the same sense, their resultant always passes through a fixed point, whose Cartesian coordinates should be obtained. [Assume that the resultant force does not vanish.]
A quadrilateral \(ABCD\) is formed from four uniform rods freely jointed at their ends. The rods \(AB\) and \(DA\) are equal in length and weight, and so also are the rods \(BC\) and \(CD\). The quadrilateral is suspended from \(A\) and a string joins \(A\) and \(C\) so that \(ABC\) is a right angle and the angle \(BAD = 2\theta\). Show that the tension in the string is \(w' + (w + w')\sin^2\theta\), where \(w\) is the weight of \(AB\) and \(w'\) is the weight of \(BC\).
Assuming that Oxford and Cambridge are 65 miles apart, and are at the same height above sea level, show that if a straight tunnel were bored between them, a train would traverse it (in one direction) under gravity alone in about 42 minutes, and find the maximum speed attained. Would your results be substantially modified if the tunnel were bored between Land's End and John O'Groats (about 600 miles apart)?