Prove that \[ \int f \frac{d^n g}{dx^n} dx = \sum_{r=1}^n (-1)^{r-1} \frac{d^{r-1}f}{dx^{r-1}}\frac{d^{n-r}g}{dx^{n-r}} + (-1)^n \int g \frac{d^n f}{dx^n} dx. \] Evaluate \[ \int_a^\infty x^m e^{-x} dx \quad \text{and} \quad \int_{-\infty}^\infty x^m \frac{d^n}{dx^n}(e^{-x^2}) dx, \] where \(m\) and \(n\) are positive integers. [The result \(\int_{-\infty}^\infty e^{-x^2} dx = \sqrt{\pi}\) may be quoted without proof.]
(i) Prove that \[ \begin{vmatrix} a_1+x & a_1 & \dots & a_1 \\ a_2 & a_2+x & \dots & a_2 \\ \vdots & \vdots & \ddots & \vdots \\ a_n & a_n & \dots & a_n+x \end{vmatrix} = x^{n-1}(x+a_1+a_2+\dots+a_n). \] (ii) Prove that \[ \begin{vmatrix} a_1 & b_1 & c_1 & d_1 \\ a_2 & b_2 & c_2 & d_2 \\ a_3 & b_3 & c_3 & d_3 \\ a_4 & b_4 & c_4 & d_4 \end{vmatrix} = \begin{vmatrix} d_4 & d_3 & d_2 & d_1 \\ c_4 & c_3 & c_2 & c_1 \\ b_4 & b_3 & b_2 & b_1 \\ a_4 & a_3 & a_2 & a_1 \end{vmatrix}. \] If \(a, b, c, d\) are real numbers, and \(p, q, r, s, t, u\) are complex numbers with respective conjugate complexes \(\bar{p}, \bar{q}, \bar{r}, \bar{s}, \bar{t}, \bar{u}\), show that all the coefficients of the polynomial in \(x\) \[ \begin{vmatrix} r-x & q & p & a \\ t & s-x & \bar{p} & b \\ u & c & \bar{s}-x & \bar{q} \\ d & \bar{u} & \bar{t} & \bar{r}-x \end{vmatrix} \] are real.
The expansion of \((1-2xy+y^2)^{-\frac{1}{2}}\) as a power series in \(y\) defines a sequence \(\{P_n(x)\}\) of polynomials in \(x\) through the identity \[ \frac{1}{\sqrt{(1-2xy+y^2)}} = \sum_{n=0}^\infty P_n(x) y^n. \] By differentiating the identity with respect to \(y\), show that \[ (n+1)P_{n+1}(x) - (2n+1)xP_n(x) + nP_{n-1}(x) = 0; \] and by differentiating the identity with respect to \(x\), show that \[ P'_{n+1}(x) - xP'_n(x) = (n+1)P_n(x). \] Show also that \[ (1-x^2)P''_n(x) - 2xP'_n(x) + n(n+1)P_n(x) = 0. \] [It may be assumed that term by term differentiation of the series is permissible.]
If \[ f(x) = \int_0^\infty \frac{e^{-x^2t}}{1+t} dt \quad (x\neq 0), \] establish the inequalities \[ f(x) < \frac{1}{x^2}, \quad \text{and} \quad f(x) < \frac{1}{1-x^2}\log\frac{1}{x^2} \quad (x \neq 1). \] [Hint. For the second inequality use \(e^{-\alpha} \le 1/(1+\alpha)\) for \(\alpha \ge 0\).] Which inequality is the stronger (i) when \(|x|\) is very large, (ii) when \(|x|\) is very small? Prove that \[ x^2f(x) \to 1 \quad \text{as} \quad |x| \to \infty. \] Show that \[ e^{-x^2}f(x) = \int_{x^2}^1 \frac{du}{u} - \int_{x^2}^1 \frac{1-e^{-u}}{u} du + \int_1^\infty \frac{e^{-u}}{u} du; \] and deduce that \[ \frac{f(x)}{\log(1/x^2)} \to 1 \quad \text{as} \quad x \to 0. \]
\(P\) is a point \((ap^2, 2ap)\) on the parabola \(y^2=4ax\). The tangents from \(P\) to the ellipse \((x-b)^2/l^2+y^2/m^2=1\) meet the parabola again at \(Q_1(aq_1^2, 2aq_1)\) and \(Q_2(aq_2^2, 2aq_2)\). Show that \(q_1, q_2\) are the roots of the quadratic equation for \(q\) \[ m^2(p+q)^2 = 4(apq+b+l)(apq+b-l). \] Verify that if \[ m^2 = 4a(b-l), \] then \(Q_1Q_2\) itself touches the ellipse, for every value of \(p\).
A curve is given in homogeneous coordinates by the parametric equations \[ x=t^3-3t, \quad y=t^2+t-4, \quad z=2t-3. \] \(P_1, P_2\) are two points on the curve with respective parameters \(t_1, t_2\). Show that \(P_1P_2\) meets the curve again at the point \(P_3\) with parameter \[ t_3 = \frac{3t_1t_2-5(t_1+t_2)+9}{2t_1t_2-3(t_1+t_2)+5}. \] Find the values of \(t_1\) and \(t_2\) for which \(t_3\) is indeterminate, and explain the geometrical significance of the indeterminacy. Show, further, that the tangents to the curve at \(P_1\) and \(P_2\) intersect at a point on the curve if \[ 2t_1t_2 - 3(t_1+t_2)+4 = 0, \] and interpret the result obtained by putting \(t_2\) equal to \(t_1\).
Two equal circular cylinders of radius \(r\) lie fixed with their axes parallel at distance \(d\) apart in a horizontal plane. A smooth prism, whose right section is a square of side \(2l\), is placed with adjacent faces resting on the two cylinders, its axis being parallel to those of the cylinders. Find, in terms of \(l\) and \(r\), the greatest value of \(d\) for which no unsymmetrical configuration of equilibrium can exist. Show that when an unsymmetrical configuration exists it is unstable. Examine the stability of the prism in its symmetrical position.
A bead of unit mass slides on a rough wire in the form of a circle of radius \(a\) whose plane is vertical; \(\theta\) is the angle between the radius to the bead and the downward vertical. Prove that \[ a\ddot{\theta} = -g(\mu\cos\theta+\sin\theta) - \mu a \dot{\theta}^2 \] when \(0 < \theta < \frac{1}{2}\pi\) and \(\dot{\theta}>0\), \(\mu\) being the coefficient of friction between bead and wire. Given that \(\mu=\frac{1}{2}\) and that the motion starts with \(\dot{\theta}=\omega\) at \(\theta=0\), find \(\dot{\theta}\) as a function of \(\theta\) during the subsequent upward motion. Find also the rate of dissipation of energy at the start. In what way, if any, is the above equation changed in the cases
Two particles, \(A\) and \(B\), of mass \(m\) and \(2m\) respectively, are connected by a light rod of length \(3a\); \(A\) is held fixed and \(B\) hangs in equilibrium. At \(t=0\), \(A\) is projected horizontally with velocity \(V\). Investigate the subsequent motion of the system. Find the depth of the particles below the original position of \(A\) when \(AB\) first becomes horizontal, and determine the kinetic and potential energies of the system at that instant, taking the value of the potential energy at \(t=0\) as standard. Explain qualitatively how the motion would be altered if the rod were replaced by a light spring that remained straight during the motion, and obtain an equation determining the maximum extension of the spring during the motion in terms of its modulus of elasticity.
A light inelastic string \(AB\) is suspended over a perfectly rough uniform pulley whose moment of inertia is \(I\) and radius \(r\). The string carries at \(A\) a mass \(m\), and at \(B\) a light spring \(BC\) of unstretched length \(a\) and elastic modulus \(\lambda\). At \(C\) is attached a second mass \(m\), and initially the system is at rest. The mass \(m\) at \(C\) is then struck with impulse \(P\) vertically downwards. Investigate the subsequent motion of the two masses.