Year: 1992
Paper: 3
Question Number: 4
Course: LFM Pure
Section: Differential equations
No solution available for this problem.
Difficulty Rating: 1700.0
Difficulty Comparisons: 0
Banger Rating: 1500.0
Banger Comparisons: 0
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.