℗ Assumematerial densitypointthinwiref(x, y, z)= €* + y2 %m. If thoontheais given bywithmassof0² t ≤ 21Ttheatonyis given bywird'sshappr(t) = thefindthe totalwire(Lino integral...)

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider the plate bounded by y² = 8x and x = 2 with density d = 2 - x. (a) Sketch a graph of the plate. Shade in the region and number the axes so that the important points on the plate can be read from the graph. (b) Express the mass of the plate as a double integral and evaluate to find the mass. (c) Express the x-coordinate of the center of mass of the plate as a double integral and evaluate to find the value of . As a side note, by the symmetry of the plate and density, the y-coordinate of the center of mass y = 0. It might be good practice to confirm this though.
ㅠ A lamina in the first quadrant is bounded by y = sin ¹ x, y = and the coordinate axes. 2' If the density at any point (x, y) in the lamina is f(x, y) = x, find the y-coordinate of the center of mass of the lamina.
6.) Carlos is sitting on a boat's deck, 18 feet above the water. When he jumps off the dock, he travels 23 feet straight down. What integer describes his new altitude?
1 %TA l. , * 0 ** h. e శికి 1:1b A 195 Partial differtial..→: JSx)dx = J =dx = In(x) Multiplying both sides of the equation by the integrating factor p(x) =x, we get x(2 ydx + xdy)= 0 = 2xydx + x'dy 0 which is exact because - 2x and = 2x, and the solution is S(x, y) = [2xydx x'y+g(y) r'y+g)-x +g'0) +g'(y) =x = g'0)-D0 g(9)-fgdy-C ry-C Mathematics Electrical & Electronic Engineering Department Sd Year Dr. Atheer AliSahr Exercises Find the solution of the following Differential Equations 2) (xdy- ydx)/ x' =0 4) 2x In(y)dx+ yr'dy 0 I) ydr + xdy =0 3) (2x+e' )dx + xe' dy = 0 5) sinh(x)cos(y)dx cosh(x)sin(y)dy 6) 3re"d0+e"dr = 0 7) (1+x')dy +2.xydx 0 8) xdy -4ydx =0 9) ydr + x(1+ y)dy =0 10) (2ydx+ dy)e" =0 11) (3y cos(3x)dx-sin(3x)dy)/ y =0 12) sin(By)dx =-ß cos(B y)dy 13) xdy - ydx =0 14) 2cos(r y)dx =r sin(7 y)dy 15) ycos(x)dx+ 3sin(x)dy = 0 16) 3ydr +2xdy =0 17) dr+(y/x)'dy =0 18) 2dx-e"dy =0 19) ycos(x)dx+ 2sin(x)dy = 0 20) (y+)dx-(x+1)dy =0 21) 2ydx +x 22) sin(y)dx + cos(y)dy = 0 23)…
Suppose the raindrop falling through the clouds accretes mass at rate dm(t)/dt yr(t)², where r(t) is the radius of the raindrop at time t with r(0) = a, y > 0, and m(t) = 4ñr³(t)p/3 with p the density of water. i) Show that r(t) = At + C and determine A and C. =

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material density at ony point thin wire is given by f(x.y₁z) = e² + y₂³%m. If the wire's shape r(t) =