Problems

Show that the equation of any circle passing through the points of intersection of the ellipse \((x+2)^2 +2y^2 =18\) and the ellipse \(9(x-1)^2 +16y^2 = 25\) can be written in the form \[ x^2-2ax +y^2 =5-4a\;. \]

Sketch the graph of the function \(\ln x - {1 \over 2} x^2\). Show that the differential equation \[ {\mathrm{d} y \over \mathrm{d} x} = {2xy \over x^2 - 1} \] describes a family of parabolas each of which passes through the points \((1,0)\) and \((-1,0)\) and has its vertex on the \(y\)-axis. Hence find the equation of the curve that passes through the point \((1,1)\) and intersects each of the above parabolas orthogonally. Sketch this curve. [Two curves intersect orthogonally if their tangents at the point of intersection are perpendicular.]

Show Solution

---

Solution:

+ - ↺

\begin{align*} && y' &= \frac{2xy}{x^2-1} \\ \Rightarrow && \int \frac{1}{y} \d y &= \int \frac{2x}{x^2-1} \d x \\ \Rightarrow && \ln |y| &= \ln |x^2-1| + C \\ \Rightarrow && y &= A(x^2-1) \end{align*} which is a family of parabolas each passing through \((\pm1, 0)\) and with a vertex on the \(y\)-axis. The curve we seek must satisfy \begin{align*} && y' &= \frac{1-x^2}{2xy} \\ \Rightarrow && \int2 y \d y &= \int \left ( \frac{1}{x} - x \right) \d x \\ \Rightarrow && y^2 &= \ln x - \tfrac12 x^2 + C \\ (1,1): && 1 &= -\tfrac12+C \\ \Rightarrow && C &= \frac32 \\ \Rightarrow && y^2 &= \tfrac32 + \ln x - \tfrac12 x^2 \end{align*}

+ - ↺

Find the two solutions of the differential equation \[ \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}=4y \] which pass through the point \((a,b^{2}),\) where \(b\neq0.\) Find two distinct points \((a_{1},1)\) and \((a_{2},1)\) such that one of the solutions through each of them also passes through the origin. Show that the graphs of these two solutions coincide and sketch their common graph, together with the other solutions through \((a_{1},1)\) and \((a_{2},1)\). Now sketch sufficient members of the family of solutions (for varying \(a\) and \(b\)) to indicate the general behaviour. Use your sketch to identify a common tangent, and comment briefly on its relevance to the differential equation.

A set of curves \(S_{1}\) is defined by the equation \[ y=\frac{x}{x-a}, \] where \(a\) is a constant which is different for different members of \(S_{1}.\) Sketch on the same axes the curves for which \(a=-2,-1,1\) and \(2\). A second of curves \(S_{2}\) is such that at each intersection between a member of \(S_{2}\) and a member of \(S_{1}\) the tangents of the intersecting curves are perpendicular. On the same axes as the already sketched members of \(S_{1},\) sketch the member of \(S_{2}\) that passes through the point \((1,-1)\). Obtain the first order differential equation for \(y\) satisfied at all points on all members of \(S_{1}\) (i.e. an equation connecting \(x,y\) and \(\mathrm{d}y/\mathrm{d}x\) which does not involve \(a\)). State the relationship between the values of \(\mathrm{d}y/\mathrm{d}x\) on two intersecting curves, one from \(S_{1}\) and one from \(S_{2},\) at their intersection. Hence show that the differential equation for the curves of \(S_{2}\) is \[ x=y(y-1)\dfrac{\mathrm{d}y}{\mathrm{d}x}. \] Find an equation for the member of \(S_{2}\) that you have sketched.