The function \(f(x,y)\) has the property that, for all \(x, y, t\), \[ f(tx,ty) = t^k f(x,y), \] where \(k\) is a constant. Prove that \begin{align*} x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} &= kf, \\ x\frac{\partial^2 f}{\partial x^2} + y\frac{\partial^2 f}{\partial x \partial y} &= (k-1)\frac{\partial f}{\partial x}. \end{align*}
Prove that, if \(x_0\) is an approximate solution of the equation \[ x \log_e x - x = k, \] and \(k_0=x_0 \log_e x_0 - x_0\), then a better approximation to the root is given by \[ x_0 + \frac{k-k_0}{\log_e x_0}. \] Given that \(\log_e 10 = 2.3026\) (to four places), find as good an approximation as you can to the root of \[ x \log_e x - x = 13. \]
Criticize the following arguments:
Obtain a reduction formula connecting the integrals \[ \int \frac{x^m \, dx}{(1+x^2)^n} \quad \text{and} \quad \int \frac{x^{m-2} \, dx}{(1+x^2)^{n-1}}. \] Prove that the value of the integral \[ \int_0^\infty \frac{x^{2k} \, dx}{(1+x^2)^{k+1}} \] decreases as the positive integer \(k\) increases. Determine the smallest (integer) value of \(k\) which makes the value of the integral less than 0.4.
Prove that, according as \(n\) is an even or odd positive integer, \[ \int_0^\pi \frac{\sin n\theta}{\sin\theta} \, d\theta = 0 \text{ or } \pi. \] If \(n\) is a positive integer, evaluate \[ \int_0^\pi \frac{\sin^2 n\theta}{\sin^2\theta} \, d\theta. \]
Determine \(P, Q, R\) as functions of \(x\) such that the equation \[ \frac{d^2y}{dx^2} + P\frac{dy}{dx} + Qy=R \] may be satisfied by \(y=x\), \(y=1\) and \(y=1/x\) for all values of \(x\) (except \(x=0\) for \(y=1/x\)). With these values of \(P, Q, R\), state what condition must be satisfied by the numerical coefficients \(a,b,c\), if the equation is also satisfied, for all \(x\) except 0, by \[ y=ax+b+\frac{c}{x}. \]
Find a polynomial of the ninth degree \(f(x)\), such that \((x-1)^5\) divides \(f(x)-1\) and \((x+1)^5\) divides \(f(x)+1\). Prove that the quotients do not vanish for any real value of \(x\).
State exactly what the statement "\(y^n e^{-y}\) tends to the limit 0 as \(y\) tends to \(+\infty\)" means. (It may be assumed true without proof.) A function \(f(x)\) of the real variable \(x\) is defined as follows: \begin{align*} f(x) &= e^{-1/x^2} \quad \text{if } x \neq 0, \\ f(x) &= 0 \quad \text{if } x = 0. \end{align*} When \(x \neq 0\), show that its \(n\)th derivative can be written \[ f^{(n)}(x) = G_n(1/x) \cdot e^{-1/x^2}, \] where \(G_n\) is a polynomial of degree \(3n\), and that the coefficients in \(G_n\) are all smaller than \(n!3^n\) in absolute magnitude. (Use mathematical induction.) Hence show that \[ |f^{(n)}(x)| < \frac{(n+1)!3^{n+1}}{|x|^{3n}} e^{-1/x^2} \] if \(0<|x|<1\). Prove that \(f^{(n)}(0)=0\) for all values of \(n\). Is there anything remarkable about this conclusion?
If \(\omega\) is one of the complex cube roots of unity, describe the position in the Argand diagram of the point \(-\omega^2z_1 - \omega z_2\). On the sides of any convex plane hexagon, equilateral triangles are constructed external to it. Their outer vertices are joined to form another hexagon. If \(PQRSTU\) are the mid-points of its sides, show that \(PS, QT\) and \(RU\) are equal and inclined at 60 degrees to one another.
A map of the world is drawn with the parallels of latitude horizontal and the meridians of longitude vertical. The parts near the equator are represented on a scale of 1 cm. to 1000 km. In the neighbourhood of every point, in order to avoid distortion, the north-and-south scale is adjusted to be equal to the east-and-west scale for that latitude. Compare the true distance from the equator of a point in latitude \(\eta\) with the distance measured on the map. [The earth may be assumed to be spherical.] A region is bounded by the meridian \(\xi=0\), the equator \(\eta=0\), and the curve \[ \xi = C(\cos\eta - \cos\beta), \] where \(\xi\) is the longitude, and \(C\) and \(\beta\) are constants less than \(\frac{1}{2}\pi\). Compare its actual area with its area on the map.