Verify that \(y = \cos x \cosh x\) satisfies the relation $$\frac{d^2y}{dx^2} = -4y.$$ Hence or otherwise show that $$y = 1 + \sum_{n=1}^{\infty} (-4)^n \frac{x^{4n}}{(4n)!}$$ \([\cosh x = \frac{1}{2}(e^x + e^{-x})]\)
(i) Evaluate $$\int_0^1 \frac{dx}{1+x^3}.$$ (ii) If \(x\) is a function of \(t\) such that $$\frac{dx}{dt} = \sqrt{\frac{x}{1-x}}$$ and \(x = 0\) when \(t = 0\) find the value of \(t\) for which \(x = 1\).
\(z = f(r)\) is a function which decreases steadily from \(h\) to \(0\) as \(r\) increases from \(0\) to \(a\). The inverse function is \(r = g(z)\). Show that $$\int_0^h [g(z)]^2 dz = 2 \int_0^a rf(r) dr$$ (i) by changing the variable in the first integral and integrating by parts; and (ii) by evaluating the volume of the solid of revolution bounded by \(z = f(r)\) and the disc \(z = 0\), \(r < a\) in two different ways. \([r = \sqrt{x^2 + y^2}; x, y, z\) are Cartesian coordinates.]
\(M(\lambda)\) is a function of the real variable \(\lambda\) defined as the greatest value of \(y = x - \lambda x^2 + \lambda x^3\) in the range \(|x| \leq 1\) of the real variable \(x\). Find the least value of \(M(\lambda)\). (ii) If \(y = x - \lambda x^2 + \lambda x^3\), for what values of \(\lambda\) is zero the least value of \(y\) in the range \(x > 0\)?
Determine the number of real positive solutions of the equation \(\log x = ax^b\) for all values of \(a\), \(b\) with \(a\) real and \(b\) real and positive.
\(z = x + iy\) and \(w = u + iv\) are complex numbers related by \(w = z^2\) and represented by points \((x, y)\) and \((u, v)\) in the \(z\) and \(w\) planes. Show that the curves in the \(w\) plane corresponding to the lines \(x = 1\) and \(y = 2\) in the \(z\) plane intersect at right angles. Comment on the fact that the curves intersect at two points.
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 \(I(x)\)?