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6. Find all extrema of the functional J(y) = 1 + (y2 + 2y) da with...
y"+ 2y' + y = 0, y(0) = 1 and y(1) = 3 Solve the initial-value differential equation y"+ 4y' + 4y = 0 subject to the initial conditions y(0) = 2 and y' = 1 Mathematical Physics 2 H.W.4 J."+y'-6y=0 y"+ 4y' + 4y = 0 y"+y=0 Subject to the initial conditions (0) = 2 and y'(0) = 1 y"- y = 0 Subject to the initial conditions y(0) = 2 and y'(0) = 1 y"+y'-12y = 0 Subject...
3) Solve the initial value problem. a) nie - 2x(y2 – 2y) = 0, with y(0) = 4 b) (-4y cos x + 4 sin I Cos I + sec? x)dx + (4y - 4 sin x)dy = 0, with y ) = 1
7. Evaluate the following integral by converting to polar coordinates: S], 127 (2x – y)dA, where R is the region in the first quadrant enclosed by the circle x2 + y2 = 4 and the lines x = 0 and y = x. 8. Find the surface area of the portion of the plane 3x + 2y +z = 6 that lies in the first octant. 9. Use Lagrange multipliers to maximize and minimize f(x, y) = 3x + y...
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1. Find and classify all the critical points of the function. f(x,y) - x2(y - 2) - y2 » 2. Evaluate the integral. 3. Determine the volume of the solid that is inside the cylinder x2 + y2- 16 below z-2x2 + 2y2 and above the xy - plane. 4. Determine the surface area of the portion of 2x + 3y + 6z - 9 that is in the 1st octant. » 5. Evaluate JSxz...
Find the absolute extrema of f(x, y) = x^2 + y^2 − 2x − 2y + 1 on the set D = {(x, y): 0 ≤ x ≤ 2 , 0 ≤ y ≤ 2 }
(d) The line integral [(x+y?)dx + (x2 + 2xy)dy, where the positively oriented curve C is the boundary of the region in the first quadrant determined by the graphs of x=0, y=x2 and y=1, can be converted to A 2xdydx 0 0 BJ 2 xdxdy 0 0 С -2x)dyda 00 D none of the above (e) Consider finding the maximum and minimum values of the function f(x, y) = x + y2 - 4x + 4y subject to the constraint...
Using the method of Lagrange Multipliers, the extrema of f(x,y) = x +y subject to the condition g(x,y) = 2x+4y -5 - O locates at B.x=1. 2 O x =2.y=0 OD. None of these The extrema of f(x,y) = x + y2 - 4x -6y +17, at critical point (2,3) is A. Maxima NB Minima O C. Saddle Point D. None of these
d2y d2y dy +6 da2 (h) +13y 2sin x +9y = 18x -+3 +6 dx da d2y (i) d2y (j d2 18x3 4y = 2 sin x dæ2 d2y ,dy .dy 9y 9x2 +21x - 10 dc (k) (1)2 7 + - 4y = e-4x +6 'da2 da2 d2y dy dy (m) 2 dæ2 (n) 4 7y= e 6 cos x 9y = 4e-3r dr2 dr dx d2y d2y (p*) dy + da2 dy (o* 2a COS I 2y 2...
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f(x,y) = -4x2 - y2 +16. – 2y + 1 if any. 6. Find equations of the tangent plane and the normal line to the surface xsin y + z2 - 4= 0 at the point (1,0,2). 7. Find the volume of the solid under the paraboloid 2 = 4 - 2 rer tb.