First order differential equations (integrating factor)
5
Differentiate \(\ln ( \ln x )\) with respect to \(x\).
Show that \(\ln x\) is an integrating factor for the first-order differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 1 } { x \ln x } y = 9 x ^ { 2 } , \quad x > 1$$
Hence find the solution of this differential equation, given that \(y = 4 \mathrm { e } ^ { 3 }\) when \(x = \mathrm { e }\).
(6 marks)