We begin by finding the EL equation for the auxiliary functional S[y] = = - S₁ 12 dx (xy2+ Ay - λy) = = · L dx F ( dxF(x,y,y', 2). d 0 = (Fy') - Fy (2xy') – (A – 1). dx dx Integrating gives A -λ B y' + 2 x and integrating again gives A - λ y = -x+Blogx+C, 2 77 come where C is another constant. The initial condition y(1) = 0 gives C = -(A - λ)/2. We now apply the natural boundary condition (HB pg 24) given by Fy = 0 at x = v. That is 0 = Fy | x=v = 2xy' | x=v' that is y'(v) = 0. Now A - X 0 = y' (v) = إن B v(A - λ) + hence B = 2 v Hence A - λ y(x) = K(x − 1 - vlogx) where K = 2

Q: y becomes 4. Show that dx y"-2 xy'+A y=D0

A:

Q: Find dx dy and. dt dt dy. dx x = 6tet, y = 2t + sin(t)

A:

Q: Find dy/dx and d²y/dx2 for r = t2 +t +1 and y = t2 + 2t - 1.

A: given that x=t2+t+1 and y=t2+2t-1 we have to find dydx and d2ydx2 Now,          dydx=dydtdxdt…

Q: dy =2xy with y(2) = 1. dx Find a function y = f(x) that satisfies

A: To find a function y=f(x) that satisfies dy/dx=2xy with y(2)=1

Q: Given y = f(u) and u = g(x), find functions. dy dx y = sin u, u = 3x - 4 = f'(g(x))g'(x) 11 dx =…

A: Topic = Derivative

Q: Use implicit differentiation to find dy/dx and d?y/dx?. x? + y? = 9 dy O A. x a?y y' dx2 x² + y? y3…

A:

Q: Find dy/dx at the given point for the equation. x = y3 – 5y2 + 2, (-2,1)

A: Given, given point is (-2,1)formula of derivatives are '

Q: of of For the function f(x,y) = 4x* - 3y' - 8, find and dx of %3D dx of

A:

Q: Find dx 2n S sin (P) at (?)dt y=

A: