Problems

Show that the second order differential equation \[\frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = f(x),\] with \(b\) and \(c\) constants, can be written as the pair of equations \[\frac{dp}{dx} - m_1 p = f(x),\] \[\frac{dy}{dx} - m_2 y = p,\] where \(m_1\), \(m_2\) are constants to be determined. Hence, or otherwise, find the general solution of \[\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} + y = e^{-x}\]

Define \(f_n(x) = n^2 x (1-x)e^{-nx}\) for \(0 \leq x \leq 1\), \(n = 0, 1, 2 \ldots\). Show that, for each \(x\) such that \(0 < x \leq 1\), \[\lim_{n \to \infty} f_n(x) = 0.\] Show also that \[\lim_{n \to \infty} \int_0^1 f_n(x) dx \neq \int_0^1 \lim_{n \to \infty} f_n(x) dx.\] [You may assume that \(\lim_{n \to \infty} nb^{-n} = 0\) for \(b > 1\).]

Show, by induction or otherwise, that, if \(n\) consecutive integers have arithmetic mean \(m\), then the sum of their cubes is \[mn\{m^2 + \frac{1}{4}(n^2-1)\}\] Find an expression in terms of \(m\) and \(n\) for the sum of their squares. Let \(s_1\) be the sum of \(n\) consecutive integers, \(s_2\) the sum of their squares and \(s_3\) the sum of their cubes. Prove that \[9s_2^2 \geq 8 s_1 s_3\]

Let \(G\) be the set of all \(2 \times 2\) real matrices of the form \[\begin{pmatrix} 1 & 0 \\ a & h \end{pmatrix}\] with \(h \neq 0\). Let \(A\) and \(H\) respectively be the set of all real matrices of the form \(\begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix}\) and the set of all real matrices of the form \(\begin{pmatrix} 1 & 0 \\ 0 & h \end{pmatrix}\) with \(h \neq 0\). Show that \(G\) is a group with respect to matrix multiplication, and that \(A\) and \(H\) are subgroups of \(G\). [You may assume that matrix multiplication is associative.]

1. [(i)] Show that every element of \(G\) may be written as a product of an element of \(A\) and an element of \(H\).
2. [(ii)] Prove that if \(X \in A\) and \(Y \in G\) then \(Y^{-1}XY \in A\). Prove also that if \(X_1, X_2 \in A\) and neither \(X_1\) nor \(X_2\) is the identity matrix, then there exists a matrix \(Z \in H\) such that \(Z^{-1}X_1Z = X_2\).

Using (i) and (ii), or otherwise, show that the only subgroups of \(G\) containing \(H\) are \(G\) and \(H\).

Let \(a\) and \(b\) be real numbers with \(a > 0\). Successive terms in the sequence \(\{f_n\}\) of real numbers are related by \[f_{n+1} = af_n + b\]

1. [(i)] If \(r\) is any real root of the polynomial \(x^3 - ax - b\), prove that \(f_n - r\) has the same sign for all values of \(n\).
2. [(ii)] Now suppose that \(x^3 - ax - b\) has three real roots \(r_1, r_2, r_3\) with \(r_1 < r_2 < r_3\). Prove that \(\{f_n\}\) is an increasing sequence if \(f_1 < r_1\) or \(r_2 < f_1 < r_3\) but is decreasing or constant otherwise.

Explain how complex numbers can be represented on an Argand diagram and demonstrate how to obtain from the positions of \(z_1\) and \(z_2\) in the diagram the positions of \(z_1 + z_2\) and \(z_1 z_2\). Interpret geometrically the inequality \[|z_1 + z_2| \leq |z_1| + |z_2|\] Prove that, if \(|a_i| \leq 2\) for \(i = 1, 2, \ldots n\), then the equation \[a_1 z + a_2 z^2 + \ldots + a_n z^n = 1\] has no solution with \(|z| \leq \frac{1}{4}\)

In a certain card game, a hand consists of \(n\) cards. Each card is either a Pip, a Queen or a Rubbish, and these occur independently of each other with probabilities \(p\), \(q\), \(r\) respectively. Calculate the expected number of Pips. The value of a hand is the product of the number of Pips with the number of Queens. Show that the expected value of a hand is \(n(n-1)pq\). [Hint: \((a + b + c)^m\) is the sum of all terms of the form \(\frac{m!}{r!s!t!}a^r b^s c^t\), where \(r, s, t\) are non-negative integers with \(r + s + t = m\).]

Using the inequality \(\int_a^b [f(x) + \lambda g(x)]^2 dx \geq 0\) for all \(\lambda\), where \(b > a\), show that \[\left(\int_a^b f(x)g(x) dx\right)^2 \leq \left(\int_a^b [f(x)]^2 dx\right)\left(\int_a^b [g(x)]^2 dx\right).\] Show that \(\frac{\pi}{4} < \int_0^{\pi/4} (\cos \theta)^{-1/2} d\theta < \{\frac{\pi}{4} \ln(1+\sqrt{2})\}^{1/2}\)

The function \(B(x, y)\) is defined by the equation, \[B(x, y) = \int_0^1 t^{x-1} (1-t)^{y-1} dt,\] for positive \(x\) and \(y\). Show that

1. [(i)] \(B(x, y) = B(y, x)\),
2. [(ii)] \(B(x, y) = B(x+1, y) + B(x, y+1)\),
3. [(iii)] \(xB(x, y+1) = yB(x+1, y)\).

Hence calculate \(B(x, y)\) when both \(x\) and \(y\) are positive integers.

Sketch the curve whose equation in polar coordinates is \[r = 1 - \frac{5}{6} \sin \theta.\] Find the range of real values of \(b\) for which the simultaneous equations \[(x^2 + y^2 + \frac{5}{6}y)^2 = x^2 + y^2\] \[y = b\] have a real solution.