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.