It is given that the equation $$x^2(1-x) \frac{d^2y}{dx^2} + Py = 0,$$ where \(P\), \(Q\) are functions of \(x\), is satisfied by \(y = x^2\) and \(y = x^3\). Find \(P\), \(Q\). With these values of \(P\), \(Q\), what condition must be satisfied by the numerical coefficients \(A\), \(B\) if the equation is satisfied by $$y = Ax^2 + Bx^3$$ for all values of \(x\)?
James. \(\pi\) is the most important constant in mathematics. John. No, \(e\) is. Continue the discussion.
Two numbers \(p\), \(q\) are given. It is required to form a cubic equation such that, if the roots are \(\alpha\), \(\beta\), \(\gamma\) (not necessarily distinct) then \(p\alpha + q\), \(p\beta + q\), \(p\gamma + q\) are also the roots. Find the cubic equation (i) for general values of \(p\), \(q\), (ii) when \(p = +1\), (iii) when \(p = -1\).
Sketch the curve \[y = \frac{(x-2)(x-3)}{(x-1)(x-4)}.\] Prove that \[\frac{dy}{dx} = \frac{-2(2x-5)}{(x-1)^2(x-4)^2}, \quad \frac{d^2y}{dx^2} = \frac{12(x^2-5x+7)}{(x-1)^3(x-4)^3},\] and deduce that the radius of curvature at the point where the curve is parallel to the \(x\)-axis has the value \(3^4/4^3\). Find all the points both of whose coordinates \(x\), \(y\) are integers, positive, negative or zero.
Given that \(s^2 + c^2 = 1\), prove that \[(4c^3 - 3c)^2 + (4s^3 - 3s)^2 = 1.\] Given, conversely, that \[(4c^3 - 3c)^2 + (4s^3 - 3s)^2 = 1,\] prove that \(s\), \(c\) satisfy one or other of three relations of the form \[ps^2 + qsc + rc^2 = 1,\] where \(p\), \(q\), \(r\) are numbers (not necessarily rational), to be determined.
Two circles \(\alpha\), \(\beta\) are each of unit radius and their centres \(A\), \(B\) are three units apart. The inverse of \(\alpha\) with respect to \(\beta\) is a point distant \(\frac{8}{9}\) from \(A\); the inverse of this point with respect to \(\beta\) is a point distant \(\frac{8}{9}\) from \(B\); and so on. The distances from \(B\), \(A\), \(B\), \(A\), \ldots are \[\frac{8}{9}, \frac{8}{9}, \frac{64}{81}, \ldots,\] the \(n\)th such distance being \[\frac{u_n}{a_{n+1}}.\] Establish the relation \[u_n - 3u_{n-1} + u_{n-2} = 0,\] and derive a formula for \(u_n\).
Two triangles \(ABC\), \(PQR\) are inscribed in a conic. The lines \(BC\), \(QR\) meet in \(L\); \(CP\), \(AR\) meet in \(M\), \(AQ\), \(BP\) meet in \(N\), so that (Pascal's Theorem, which you are not asked to prove) \(L\), \(M\), \(N\) lie on a line, meeting the conic in two points \(I\), \(J\), assumed distinct. The figure is such that \(L\) lies on \(AP\), \(M\) lies on \(BQ\), \(N\) lies on \(CR\). Prove that \(I\), \(J\) are the self-corresponding points of each of the (1, 1) correspondences
Prove that the equation of the chord of the conic \[ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0\] with middle point \((x_1, y_1)\) is \[(ax_1 + hy_1 + g)(x - x_1) + (hx_1 + by_1 + f)(y - y_1) = 0,\] the axes not necessarily being rectangular. A conic \(S\) and two lines \(p\), \(q\) are given, with the property that every chord \(UV\) of the conic meets the lines in two points \(P\), \(Q\) such that \(UV\), \(PQ\) have the same middle point. Determine whether \(p\), \(q\) are necessarily the asymptotes of the conic.
Show that necessary and sufficient conditions for the equilibrium of a system of coplanar forces are that the resultant forces in any two directions are zero, and that the sum of the moments of the forces about any point in the plane is zero. A uniform ladder, weight \(W\), leans against a vertical wall and makes an angle \(\theta\) with the horizontal ground. The vertical plane through the ladder is perpendicular to the wall. The coefficient of friction between the ladder and both the ground and the wall is \(\mu\). A force \(P\), perpendicular to and away from the wall, is applied to the top of the ladder. Obtain an expression in terms of \(P\) for the minimum value of \(\mu\) which will prevent the ladder from slipping. Show that if \(\mu > \frac{1}{2}\cot\theta\) it will not be possible to make the ladder slip for any value of \(P\).
A thin uniform plank, length \(2l\) and weight \(W\), rests on a fixed circular radius \(a\), whose axis is perpendicular to the length of the plank. The plank is horizontal position by a vertical string, under tension \(T_0\), attached to one end of the plank and to a point \(P\) above the plank. \(P\) is then moved so that the plank makes an angle \(\alpha\) with the horizontal, the string remaining vertical. Assuming that the plank is sufficiently long to prevent slipping, find an expression for the tension in the string. If the coefficient of friction between the plank and the cylinder, \(\mu\), is very much less than one, and maximum and minimum possible values of the tension, before slipping occurs, differ by \[2\mu(T_0 - W)^2/W].\]