The equation of a conic referred to rectangular cartesian axes is $$ax^2 + 2hxy + by^2 = 1.$$ Show that the locus of the mid-points of chords parallel to the diameter \(x/l = y/m\) is the line \((al + hm)x + (hl + bm)y = 0\) (the conjugate diameter). Prove the equivalence of the following definitions of the principal axes:
Show that the function $$f(x) = e^{-x} \int_{-\infty}^{x} e^{s} F(s) ds$$ satisfies the differential equation $$f'(x) + f(x) = F(x).$$ The function \(\phi(x)\) is defined as follows: $$\phi(x) = 0 \quad \text{for } x < 0$$ $$x \quad \text{for } x > 0$$ Given that $$f'(x) + f(x) = \phi(x) \quad \text{and} \quad f(-\infty) = 0,$$ find \(f(x)\). Show graphically the forms of the functions \(\phi(x)\) and \(f(x)\). Given that $$g'(x) + g(x) = \phi(x) - \phi(x-1) \quad \text{and} \quad g(-\infty) = 0,$$ find the function \(g(x)\) and show graphically the forms of the functions \(\phi(x) - \phi(x-1)\) and \(g(x)\).
A function \(y\) of \(x\) and \(\lambda\) is defined by the equation $$y = x^2 + \lambda x^2 y^{-\frac{1}{2}}$$ where \(\lambda\) is small. Assuming that \(y\) may be expressed in the form $$p(x) + \lambda q(x) + \lambda^2 r(x) + \lambda^3 s(x) + \ldots$$ find the functions \(p(x), q(x), r(x)\) and \(s(x)\). Convergence need not be discussed.
By considering $$\int_a^b \{f(x) + \lambda g(x)\}^2 dx$$ show that $$\left\{\int_a^b fg dx\right\}^2 \leq \int_a^b f^2 dx \cdot \int_a^b g^2 dx.$$ By applying this inequality to the integrals $$\int_0^1 (x^{\frac{1}{2}})(x^2-1) dx \quad \text{and} \quad \int_0^1 (x^{\frac{1}{2}})(x^{\frac{1}{2}} e^{-x}) dx,$$ show that $$\int_0^1 x^{\frac{1}{2}} e^x dx$$ lies between 1.11 and 1.13. \([e = 2.71828; e^2 = 7.38906; \sqrt{e} = 1.64872.]\)
A particle of mass \(m\) falls in a vertical plane from rest under the influence of constant gravitational force \(mg\) and a force \(mkv\) perpendicular to its velocity, where \(k\) is a constant. Write down expressions for \(v\) and the curvature of its path after it has fallen through a vertical distance \(y\). Show that \(y\) never exceeds \(2g/k^2\).
The mass per unit surface area of a thin spherical shell of radius \(a\) is proportional to the square of the distance from a point on its surface. It is suspended from by light rods from a fixed point at its centre. If \(k_1\) is the radius of gyration about a horizontal axis through the centre and \(k_2\) is the radius of gyration about the vertical axis through the centre, prove that $$2k_1^2 + k_2^2 = 2a^2.$$ Show that the period of small planar oscillations under gravity is the same as that of a simple pendulum of length \(2a\).
A light rod \(OA\) of length \(l\) rotates freely about a fixed point \(O\). A point particle of mass \(m\) attached to the rod at \(A\) is initially at rest vertically below \(O\). A projectile of mass \(m\) moving horizontally with speed \(v\) (\(v^2 < 16gl\)) embeds itself instantaneously in the target. Obtain the height \(h\) through which the target would rise before first coming to rest if undisturbed in the subsequent motion. However, after rising through a height \(3h/4\) another similar projectile embeds itself in the target. How much further will the target rise? If the total height through which the target rises is \(3h/4 + h'\), show that \(h'\) is greatest (for variable \(v\)) if \(v^2 = 16gl/3\).
If $$f(x) = \frac{(1+x)^{\frac{1}{2}} - 1}{1-(1-x)^{\frac{1}{2}}},$$ find (i) \(\lim_{x \to 0} f(x)\), and (ii) \(\lim_{x \to 0} \frac{df}{dx}\).
If \(y = (x^2-1)^n\), where \(n\) is a positive integer, prove that $$(1-x^2)\frac{dy}{dx} + 2nxy = 0.$$ By differentiating this equation \((n+1)\) times and using Leibniz' theorem, or otherwise, show that the function \(p_n(x)\), defined by $$p_n(x) = \frac{d^n}{dx^n}(x^2-1)^n,$$ satisfies the equation $$(1-x^2)\frac{d^2p_n}{dx^2} - 2x\frac{dp_n}{dx} + n(n+1)p_n = 0.$$ Show also that $$p_n(1) = (-1)^n p_n(-1) = 2^n n!$$
Find the minimum distance between the origin and the branch of the curve $$y = \frac{x}{x+c} \quad (c \neq 0)$$ that does not pass through the origin. Does your result remain true when \(c = 0\)?