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.
Practise Edexcel A Level Maths Integration with exam-style questions for A Level Maths. 437 questions 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, matched to the Edexcel A Level Maths (9MA0) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
