Obtain the complete solution of the equation \[ x\frac{d^2y}{dx^2} + (\gamma+1)\frac{dy}{dx}-xy=0, \] in the form of series of ascending powers of \(x\), \(\gamma\) being any real number.
Give an account of Lagrange's method of solving the linear partial differential equation \[ Pp+Qq=R, \] explaining the geometrical interpretation of the method. Solve the equation \[ yp-xq=c \quad (c>0). \] Describe the general nature of the surfaces represented by this equation.
A function \(f(z)\) is regular (holomorphic) in the domain \(D\) obtained by excluding from the \(z\)-plane the two regions defined by
\begin{align*}
x < -\frac{1}{2}-\delta, \quad y \ge 0 \\
x \ge +\frac{1}{2}+\delta, \quad y \le 0
\end{align*}
(\(z=x+iy\)) (\(\delta\) being a positive constant), and satisfies throughout \(D\) the inequality
\[
|f(z)|
Prove the addition formula \[ \wp(u+v) = \frac{1}{4}\left(\frac{\wp'u-\wp'v}{\wp u-\wp v}\right)^2 - \wp u - \wp v \] for the Weierstrassian elliptic function \(\wp u\), and deduce a formula for \(\wp(2u)\). Shew that if \(\wp u\) has primitive periods \(2\omega_1, 2\omega_2\), and invariants \(g_2, g_3\), then \(\wp\left(\frac{2\omega_1}{3}\right)\) is a root of the equation \[ 48x^4-24g_2x^2-48g_3x-g_2^2=0. \] Find the other roots of this equation.
If \(f(x) = (x+1)(2x^2-x+1)^{1/2}(x-1)^{-1/2}\) prove that \(f(x) = f(\{1-x\}/\{1+x\})\). Show that the equation \(f(x) = f(3)\) has two real and two imaginary roots, giving the values of each.
Prove that the product of any set of integers, each of which can be expressed as the sum of the squares of two integers, is equal to the sum of the squares of two integers. Express 7540 as the sum of the squares of two integers.
Show that, if \(a_1, a_2, \dots, a_m\) are distinct prime numbers other than unity, the number of solutions in integers (including unity) of the equation \(x_1 x_2 x_3 \dots x_n = a_1 a_2 a_3 \dots a_m\) is \(n^m\). Show also that the number of solutions in which at least one of the \(x\)'s is unity is \[ n! \left\{ \frac{(n-1)^m}{(n-1)! 1!} - \frac{(n-2)^m}{(n-2)! 2!} + \dots + (-)^{n-2} \frac{1^m}{1!(n-1)!} \right\}. \]
Find the sum of the terms after the \(n\)th in the expansion of \((1+x)/(1-x)^2\) in ascending powers of \(x\). Prove that the ratio of this sum to the sum of the corresponding terms in the expansion of \((1-x)^{-2}\) can be made equal to any given number \(\lambda\), which is greater than \(2n\), by suitable choice of \(x\). Explain clearly why the restriction upon \(\lambda\) is necessary.
In any triangle \(ABC\) prove that the sum of the squares of the distances of the centre of the inscribed circle from the vertices is \(bc+ca+ab - 6abc/(a+b+c)\). Investigate the corresponding result for the sum of the squares of the distances of the centre of an escribed circle from the vertices.
Prove that the least value of \(a\cos\theta + b\sin\theta\) is the negative square root of \(a^2+b^2\). Prove also that the least value of \[ x^2 + 2x(a\cos\theta+b\sin\theta) + c\cos 2\theta + d\sin 2\theta \] is \[ -\frac{1}{2}(a^2+b^2) - \{c^2+d^2+\frac{1}{4}(a^2+b^2)^2+c(b^2-a^2)-2abd\}^{1/2}. \]