List of top Differential Equations Questions

Let \( f : [0, \infty) \to [0, \infty) \) be continuous on \( [0, \infty) \) and differentiable on \( (0, \infty) \). If \[ f(x) = \int_0^x \sqrt{f(t)} \, dt, \text{ then } f(6) = \, \text{..........} \]

If \( x(t) \) is the solution to the differential equation \( \frac{dx}{dt} = x^2 t^3 + x t, \text{ for } t > 0, \text{ satisfying } x(0) = 1, \) then the value of \( x(\sqrt{2}) \) is .......... (correct up to two decimal places). 
 

If \( y(x) = v(x) \sec x \) is the solution of \[ y'' - (2 \tan x) y' + 5y = 0, -\frac{\pi}{2} < x < \frac{\pi}{2}, \text{ satisfying } y(0) = 0 \text{ and } y'(0) = \sqrt{6}, \] then \( v \left( \frac{\pi}{6 \sqrt{6}} \right) \) is .............. (correct up to two decimal places).