Assuming that the equation \[ x\frac{d^2y}{dx^2} + \frac{dy}{dx} - m^2xy = 0 \] is satisfied by a solution of the form \[ y = \sum_{r=0}^{\infty} c_r x^{\alpha+r}, \] where \(c_0\) is not zero, find the value of \(\alpha\), prove that \(c_{2r+1}=0\) and that \[ 4r^2c_{2r} = m^2c_{2r-2}, \] and write down the first four terms of the series.
Shew that the equation \[ x^2+y^2+2gx+c=0 \] represents, for a given \(c\) and different \(g\)'s, a system of coaxal circles. Prove that the poles of a fixed line \(l\) with respect to the circles of a coaxal system lie on a fixed conic \(S\). Shew that, if \(l\) is not parallel to the radical axis or to the line of centres and does not pass through a limiting point of the system, then \(S\) is a (non-degenerate) hyperbola whose asymptotes are given by the following construction. Let \(l\) meet the line of centres and the radical axis in \(P\) and \(Q\) respectively; take \(R\) on \(l\) so that \(QR=PQ\); then the asymptotes are the lines through \(P\) and \(R\) perpendicular to \(l\) and parallel to the radical axis respectively.
A horse pulls a cart starting from rest at \(A\); the pull exerted gradually decreases until on reaching \(B\) it is equal to the constant resistance to motion due to friction, etc. Shew that, if the decrease in pull is proportional to the time from leaving \(A\), the velocity at \(B\) is \(\frac{\sqrt{3}}{2}\) of the velocity at \(B\) if the decrease is proportional to the distance from \(A\), the initial pull being the same in each case.
A uniform sphere of weight \(W\) rests on a horizontal plane touching it at \(C\). A uniform beam \(AB\) of weight \(X\) has its end \(A\) on the plane and is a tangent to the sphere at \(B\), \(ABC\) being a vertical plane. The ratios of the tangential to the normal reactions at \(A,B,C\) in equilibrium are \(\alpha, \beta, \gamma\). Shew that \(\beta\) is greater than \(\alpha\) and than \(\gamma\), and that \(\alpha > \gamma\) if \(W > X(1-\cos BAC)\).
Prove that, if \[ x=r \sin\theta \cos\phi, \quad y=r \sin\theta \sin\phi \quad \text{and} \quad z=r \cos\theta, \] then \[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right)^n \frac{1}{r^m} = \frac{(m-1)m(m+1)(m+2)\dots(m+2n-2)}{r^{m+2n}}, \] and find the value of \[ \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right) \log \tan \frac{1}{2}\theta. \]
The lines joining a point \(P\) on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) to the points \((\pm c, 0)\) meet the ellipse again in \(Q\) and \(R\). The tangents at \(Q\) and \(R\) meet in \(T\). Prove that \(PT\) is bisected by the minor axis.
A gun and an object fired at are in a horizontal plane, and the angle of elevation necessary to hit the object is \(\theta \left(<\frac{\pi}{4}\right)\). If the elevation is increased by \(\alpha\) the shot passes through a point at a vertical distance \(k\) from the object; if the vertical plane of projection is rotated through the same angle \(\alpha\) about the vertical through the gun the shot hits the plane at a distance \(c\) from the object. Find the value of \(\frac{k}{c}\) in terms of \(\theta\) and \(\alpha\), and shew that, if \(\alpha\) is small, \[ k=c(1-\tan^2\theta - 2\alpha \tan^3\theta), \text{ approximately}. \]
A particle of mass \(m\) is placed on the centre of a plank of length \(2l\) and mass \(M\) which rests on a horizontal table. A horizontal impulse is then applied to the plank in the direction of its length. If the coefficients of friction between the plank and the table and between the plank and the particle are both \(\mu\), shew that the magnitude of the impulse which just suffices to knock the plank clear of the particle is \[ \{4\mu glM(M+m)\}^{\frac{1}{2}}. \]
Prove that, if \(r, r'\) denote the distances of a point \(P\) from two fixed points \(A, B\) and \(\theta, \theta'\) the angles that \(AP, BP\) make with a fixed direction, then the families of curves \[ r^m r'^n = \alpha, \] \[ m\theta + n\theta' = \beta, \] where \(m\) and \(n\) are constants and \(\alpha, \beta\) are variable parameters, cut one another orthogonally.
\(AB\) is a fixed diameter of a rectangular hyperbola and \(P\) a variable point on the hyperbola. Prove that the difference between the angles \(PAB\) and \(PBA\) is equal to one of two fixed supplementary angles. Hence or otherwise prove that the two circles touching the hyperbola at \(P\) and passing respectively through \(A\) and \(B\) are of equal radius.