First order differential equations (integrating factor)
3
Show that \(y = x ^ { 3 } - x\) is a particular integral of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$
By differentiating \(\left( x ^ { 2 } - 1 \right) y = c\) implicitly, where \(y\) is a function of \(x\) and \(c\) is a constant, show that \(y = \frac { c } { x ^ { 2 } - 1 }\) is a solution of the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 0$$
Hence find the general solution of
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 x y } { x ^ { 2 } - 1 } = 5 x ^ { 2 } - 1$$