First order differential equations (integrating factor)
7
Given that \(y = \ln ( \ln x )\), show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln x }$$
The variables \(x\) and \(t\) satisfy the differential equation
$$x \ln x + t \frac { \mathrm {~d} x } { \mathrm {~d} t } = 0$$
It is given that \(x = \mathrm { e }\) when \(t = 2\).
Solve the differential equation obtaining an expression for \(x\) in terms of \(t\), simplifying your answer.
Hence state what happens to the value of \(x\) as \(t\) tends to infinity.