Homogeneous differential equations: notes and practice questions

This question explores families of curves and their intersections, including orthogonal trajectories and curves intersecting at a specific acute angle.

Consider a family of curves, $L$, with equation $xy = c$, where $c$ is a parameter. Each member of $L$ intersects every member of a family of curves, $C$, at right-angles.

Note: In parts (i), (ii) and (iii), you are not required to consider the case where $x = 0$ or $y = 0$.

Write down an expression for the gradient of $L$ in terms of $x$ and $y$.

Hence show that the gradient of $C$ is given by $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x}{y}$.

By solving the differential equation $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x}{y}$, show that the family of curves, $C$, has equation $y^2 - x^2 = K$ where $K$ is a parameter.

Consider two families of curves: $F_1$ with equation $x^2 - y^2 = A$ and $F_2$ with equation $xy = B$. For this part, let $A = 3$ and $B = 2$.

On the same set of axes, sketch the curves $x^2 - y^2 = 3$ and $xy = 2$. On your sketch, clearly label each curve and any $x$-intercepts.

Find the coordinates of the intersection points of the curves $x^2 - y^2 = 3$ and $xy = 2$.

At the point $(2, 1)$, show that the curves $x^2 - y^2 = 3$ and $xy = 2$ intersect at right-angles.

Consider two families of curves, $F$ and $G$.

The gradient of $F$ is denoted by $f(x, y)$.

The gradient of $G$ is denoted by $g(x, y)$.

Each member of $F$ intersects every member of $G$ at an acute angle, $\alpha$.

It can be shown that

$g(x, y) = \frac{f(x, y) + \tan \alpha}{1 - f(x, y) \tan \alpha}$

In part (e), consider the specific case where $f(x, y) = -\frac{y}{x}$, for $x \neq 0$, $y \neq 0$ and $\alpha = \frac{\pi}{4}$.

Show that $g(x, y) = \frac{x-y}{x+y}$.

Hence, by solving the homogeneous differential equation $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x-y}{x+y}$, find a general equation that represents this family of curves, $G$. Give your answer in the form $h(x, y) = d$ where $d$ is a parameter.

By considering $\lim_{\alpha \to \frac{\pi}{2}} \tan \alpha$, show that, for all finite $f(x, y)$,

$\lim_{\alpha \to \frac{\pi}{2}} g(x, y) = -\frac{1}{f(x, y)}$.