An engineer is modeling the cross-sectional profile of a specialized optical lens. The thickness of the lens, y y\,y mm, at a horizontal distance x x\,x mm from the optical axis, satisfies a specific differential equation.
Find the derivative with respect to y y\,y of
1(1+2lny)2 \frac{1}{(1 + 2\ln y)^2} (1+2lny)21Hence find a general solution to the differential equation
12csc(2x)dydx=y(1+2lny)3 12\csc(2x) \frac{dy}{dx} = y(1 + 2\ln y)^3 12csc(2x)dxdy=y(1+2lny)3for y>0 y > 0\,y>0 and −π2<x<π2\displaystyle -\frac{\pi}{2} < x < \frac{\pi}{2}−2π<x<2π.
Show that the particular solution of this differential equation for which y=e1/2y = e^{1/2}y=e1/2 at x=π6\displaystyle x = \frac{\pi}{6}x=6π is given by
y=eAsecx−12 y = e^{A\sec x - \frac{1}{2}} y=eAsecx−21where A A\,A is a constant to be found.
864 exam-style questions on Edexcel A Level Maths Integration, covering 11.1 Integrating Standard Functions, 11.2 Integrating f(ax + b), 11.3 Using Trigonometric Identities, 11.4 Reverse Chain Rule, 11.5 Integration by Substitution, 11.6 Integration by Parts, 11.7 Partial Fractions, 11.8 Finding Areas, 11.9 The Trapezium Rule, 11.10 Solving Differential Equations, and 11.11 Modelling with Differential Equations. Each one has a worked solution and a mark scheme showing where the marks go.