An engineer is modeling the cross-sectional profile of a specialized optical lens. The thickness of the lens, $y\,$ mm, at a horizontal distance $x\,$ mm from the optical axis, satisfies a specific differential equation.

Find the derivative with respect to $y\,$ of

\frac{1}{(1 + 2\ln y)^2}

Hence find a general solution to the differential equation

12\csc(2x) \frac{dy}{dx} = y(1 + 2\ln y)^3

for $0\,$ and $−π2<x<π2\displaystyle -\frac{\pi}{2} < x < \frac{\pi}{2}$ .

Show that the particular solution of this differential equation for which $y = e^{1/2}$ at $x=π6\displaystyle x = \frac{\pi}{6}$ is given by

e^{A\sec x - \frac{1}{2}}

where $A\,$ is a constant to be found.

Find the derivative with respect to $y\,$ of

\frac{1}{(1 + 2\ln y)^2}

Hence find a general solution to the differential equation

12\csc(2x) \frac{dy}{dx} = y(1 + 2\ln y)^3

for $0\,$ and $−π2<x<π2\displaystyle -\frac{\pi}{2} < x < \frac{\pi}{2}$ .

Show that the particular solution of this differential equation for which $y = e^{1/2}$ at $x=π6\displaystyle x = \frac{\pi}{6}$ is given by

e^{A\sec x - \frac{1}{2}}

where $A\,$ is a constant to be found.

Integration

Question 48

Markscheme

Integration

Question 48

Markscheme

Integration

Question 48

Integration Questions

Integration

Question 48

Integration Questions