Solve the differential equation dr/dx = ln x / x² for x > 0 given x = 1 when t = 2 Write your answer in the form t² = f(x) - AQA - A-Level Maths Mechanics - Question 5 - 2019 - Paper 2

Next, we integrate both sides. We need to integrate the right side:

$$$$

Given the condition x = 1 when t = 2, we substitute into the equation:

$t^2 = -\frac{ ext{ln} (1)}{1} + ext{ln} (1) + C$

Since ln(1) = 0, we have:

$2^2 = 0 + 0 + C \\ C = 4$

Now substitute the value of C back into the equation:

$t^2 = -\frac{ ext{ln} x}{x} + ext{ln} x + 4$

To make it clearer, the final form is:

$t^2 = -\frac{ ext{ln} x}{x} + ext{ln} x + 4$

This can be further simplified to:

$t^2 = 4 + ext{ln} x \left(1 - \frac{1}{x}\right)$