11 In a field there are 300 plants of a certain species, all of which can be infected by a particular disease. At time \(t\) after the first plant is infected there are \(x\) infected plants. The rate of change of \(x\) is proportional to the product of the number of plants infected and the number of plants that are not yet infected. The variables \(x\) and \(t\) are treated as continuous, and it is given that \(\frac { \mathrm { dx } } { \mathrm { dt } } = 0.2\) and \(x = 1\) when \(t = 0\).

1. Show that \(x\) and \(t\) satisfy the differential equation $$1495 \frac { \mathrm { dx } } { \mathrm { dt } } = x ( 300 - x )$$
2. Using partial fractions, solve the differential equation and obtain an expression for \(t\) in terms of a single logarithm involving \(x\).
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