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.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.