MATH 2650 Calculus of Variations Examples 1 The practice ...

MATH 2650 Calculus of Variations Examples 1The practice questions (marked P) will be discussed in the Examples Classes to be held in Weeks2 and 3. Answers to questions Q1 – Q11 are to be handed in on Wednesday 17 October 2012.P1) Find the general solution of the first order ODEs:(a) dydx + y x= sin x, (b) x2dydx + xy = y2 , (c) dydx = x(1 + x2 ) 1/2 tan y, (d) dydx = x2 + xy + y 2x 2 .P2) Find the general solution of the second order ODE:d 2 ydx + dy2 dx − 2y = f(x), when (a) f(x) = x + 3, (b) f(x) = sin x, (c) f(x) = ex .P3) f(x, y) = x/(x + y). Calculate ∂ x f and ∂ y f. Then show that ∂ 2 xyf = ∂ 2 yxf.P4) Find ∂ x f and ∂ z f iff(x, y, z, w) = (xw) yz .P5) Find all second order partial derivatives of f(x, y) = x xy .P6) Using the chain rule, calculate df/dx ifP7) Using the chain rule, calculate df/dt iff(x, y) = 1/ ln(xy) and y = sin x.f(x, y, t) = t 2 + xyt, with x = cos t, y = sin t.P8) Using a Taylor series, express x 3 − 5x 2 + 3x − 6 as a polynomial in (x − 2).P9) Use the method of Lagrange multipliers to find maximum and minimum values of thefunction f(x, y) subject to the constraint g(x, y) for the following cases:(a) f(x, y) = x 2 + y 2 , g(x, y) = 2x + 3y = 18;(b) f(x, y) = xy, g(x, y) = x 4 + y 4 = 2.P10) Consider a cuboidal coal bunker against a wall. For a given volume V , what dimensionsx, y, z minimise the exposed surface area?P11) By differentiating under the integral sign, calculate the following integral:I(x) =∫ 10t x − 1ln t dt.P12) Find the extremals (ignoring boundary conditions) offor each of the cases:(a) F = y ′2 − k 2 y 2 ,(b) F = xy ′2 − yy ′ + y.I =∫ x2x 1F (x, y, y ′ )dx

Q1) Find the general solution of the first order ODEs:(a) y − 2x dydx = x(x + 1)y3 , (b) dydy+ y tan x = sin x, (c)dx dx = x2 e x3 y 2 ,(d) (x 3 +y 3 ) = 3xy 2 dy , (e) x(x+1)dydx dx = y(x−1).Q2) Find the general solution of the second order ODEs:(a) y ′′ +3y ′ +2y = e 4x , (b) y ′′ +4y ′ +3y = cos 3x, (c) y ′′ +2y ′ +2y = e x sin 2x,(d) y ′′ + y = 3e x + e 2x , (e) y ′′ + 25y = sin 5x, (f) y ′′ − 2y ′ − 3y = 2e 3x .Q3) Find all second order partial derivatives of f(x, y) = xe xy .Q4) Using the chain rule, calculate du/dt (in terms of t) ifu(x, y, t) = t 2 x + xyt + x 3 , x = t 2 + 3, y = t 3 .Q5) Using a Taylor series, express the following as polynomials in (x − 3):(a) x 2 − 10x + 7, (b) x 5 .Q6) Use the method of Lagrange multipliers to find maximum and minimum values of thefunction f(x, y) subject to the constraint g(x, y) for the following cases:(a) f(x, y) = xy, g(x, y) = x 2 + y 2 = 18;(b) f(x, y) = x + 2y, g(x, y) = x 2 + 2y 2 = 12.Q7) What is the largest rectangle that can be inscribed in the ellipse x 2 + 2y 2 = 1?Q8) Calculate dI/dx, where I(x) is defined as the following integral:I(x) =∫ x 2x1t ext dt.Q9) Find the extremals (ignoring boundary conditions) of I = ∫ x 2x 1F (x, y, y ′ )dx for each of thecases:(a) F = y ′2 +2y, (b) F = y ′2 +4xy ′ , (c) F = y ′2 +yy ′ +y 2 .Q10) Find the extremals of:(a)(b)∫ 10∫ 10(y 2 + y ′ + y ′2) dx subject to y(0) = 0, y(1) = 1;y ′2dx subject to y(0) = 1, y(1) = 2.x3 Q11) Show that, if y(x) satisfies the Euler-Lagrange equation associated with the integral∫ x2(I = p 2 y ′2 + q 2 y 2) dx,where p(x) and q(x) are given functions, then I takes the valuex 1I = [ p 2 yy ′] x 2x 1.