Find:
Obtain a recurrence relation connecting \(F(p)\) and \(F(p+1)\), where \(F(p) = \int_0^1 x^p (1-x)^{-1/4} dx,\) Hence, or otherwise, evaluate \(F(2)\) and \(F(\frac{3}{2})\).
Show that \(y = (x + (x^2 + 1)^{1/2})^k\) satisfies the differential equation \((x^2 + 1)y'' + xy' - k^2y = 0,\) Derive an equation connecting \(y^{(n)}(x)\), \(y^{(n+1)}(x)\) and \(y^{(n+2)}(x)\). Hence show that, if \(k\) is an integer, then \(y^{(k+1)}(x) = A(x^2 + 1)^{-k-\frac{1}{2}},\) where \(A\) is a constant, and find \(A\).
Prove that, if \(y > x > 0\) and \(k > 0\), then \(x^k (y-x) < \int_x^y t^k dt < y^k (y-x).\) Hence show that \(\lim_{n \to \infty} \{n^{k-1}(1^k + 2^k + \ldots + n^k)\} = \frac{1}{k+1}.\)
Show that, if \(u = r + x\), \(v = r - x\), where \(r = (x^2 + y^2)^{1/2}\), and \(f(x,y) = g(u,v)\), then \(r\left(\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}\right) = 2u \frac{\partial^2 g}{\partial u^2} + 2v \frac{\partial^2 g}{\partial v^2} + \frac{\partial g}{\partial u} + \frac{\partial g}{\partial v}.\)
Prove that the solution of the differential equation \(\frac{dy}{dx} + ay = f(x),\) where \(a\) is constant, is \(y = y_0 e^{-ax} + \int_0^x f(u) e^{a(u-x)} du,\) where \(y_0 = y(0)\). Hence, or otherwise, solve \(\frac{d^2 y}{dx^2} + (a+b) \frac{dy}{dx} + aby = \begin{cases} 1, & (0 < x < 1) \\ 0, & (x > 1), \end{cases}\) where \(a\) and \(b\) are constants \((a \neq b)\), given that \(y = 0\), \(dy/dx = 0\) for \(x = 0\), and that \(y\) and \(dy/dx\) are continuous.
If, for all \(x\) such that \(0 \leq x \leq h\) (\(h > 0\)), $$|c_0 + c_1x + c_2x^2 + \ldots + c_nx^n| \leq Ax^{n+1}$$ where \(A\) is a constant and \(n\) a given positive integer), show that the constants \(c_0, c_1, \ldots, c_n\) are all zero. A function \(f(x)\) is differentiable as many times as we wish, and in the interval \(0 \leq x \leq h\) its \((n-1)\)th derivative \(f^{(n-1)}(x)\) lies between \(\pm K_n\), for each \(n\). Prove by successive integration that, for \(0 \leq x \leq h\), $$f(x) - f(0) - xf'(0) - \ldots - \frac{x^n}{n!}f^{(n)}(0)$$ is between \(\pm K_n x^{n+1}/(n+1)!\). Deduce that if \(g(x) \equiv f(x^2)\) then \(g^{(2p+1)}(0) = 0\), \(g^{(2p)}(0) = (2p)!f^{(p)}(0)/p!\) for every positive integer \(p\). [It must not be assumed that the infinite Taylor series of \(f(x)\) converges to sum \(f(x)\).] Without assuming the binomial theorem for fractional indices, find the sixth derivative at \(x = 0\) of \((1-x^2)^{1/2}\).
Sketch the family of curves $$(x-a)^2 - y^2 + y^3 = 0,$$ where \(a\) is a parameter. Show that the usual method of finding an envelope, by eliminating \(a\) between the equations \(F(x, y, a) = 0\) and \(\partial F/\partial a = 0\), yields in this case not only the envelope but also an extraneous locus. How do you account for this? Prove that for a family of straight lines of the form \(y = ax + P(a)\), where \(P\) is a polynomial of degree at least two, the standard method always yields a true envelope (that is, a curve whose slope at any point is the same as that of some straight line of the family which passes through that point). Find, in the form \(\phi(x, y) = 0\), the envelope of the family of lines \(y = ax + 3x - a^3\), and sketch it.
Show that, if \(P(x)\) is a polynomial of degree \(n\) such that the \(n\) repeated factors, then between any two consecutive real roots of the equation \(P(x) = 0\), the polynomial function must possess a maximum or minimum between any two consecutive real zeros of the function.) Show that, if the equation \(P(x) = 0\) has \(n\) real roots (counting multiple roots according to their multiplicity), then \(P'(x) = 0\) has \(n-1\) real roots. If \(P(x) = 0\) has \(m-1\) real roots, all distinct, where \(m < n\), show that \(P(x) = 0\) has \(m\) or \(m-2k\) real roots, where \(k\) is some positive integer. Show that this remains true if one, but only one, of the real roots of \(P'(x) = 0\) is repeated. A sequence of functions \(Q_n(x)\) is defined by the relations $$Q_0(x) = 1 \quad (\text{all } x),$$ $$Q_n(x) = xQ_{n-1}(x) - Q'_{n-1}(x) \quad (n \geq 1).$$ Prove that \(Q_n(x)\) is a polynomial of leading term \(x^n\) and with \(n\) distinct real zeros.
Obtain a reduction formula for $$I_n = \int \frac{x^n dx}{(ax^2 + c)^{1/2}}$$ where \(a\), \(c\) are real non-zero constants. Show that by the use of this formula (if necessary, with \(n\) negative), \(I_m\) can, for any positive or negative integer \(m\), be expressed in terms of known functions and \(I_0\) or \(I_{-1}\) according to the sign of \(a\) and \(c\). Obtain explicitly $$\int \frac{x^4 + 1}{(x^2 + 4)^{1/2}} dx \quad \text{and} \quad \int \frac{dx}{x^3(1-4x^2)^{1/2}}.$$