4 The variables \(x\) and \(y\) are related by the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - y ^ { 2 } } { x y }$$
- Use the substitution \(y = x z\), where \(z\) is a function of \(x\), to obtain the differential equation
$$x \frac { \mathrm {~d} z } { \mathrm {~d} x } = \frac { 1 - 2 z ^ { 2 } } { z }$$
- Hence show by integration that the general solution of the differential equation (A) may be expressed in the form \(x ^ { 2 } \left( x ^ { 2 } - 2 y ^ { 2 } \right) = k\), where \(k\) is a constant.