MEC702_Wk_5_Tutorial

# MEC702_Wk_5_Tutorial - MEC702 Wk 5 Tutorial 1 Solve the ODE...

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Unformatted text preview: MEC702 Wk 5 Tutorial 1. Solve the ODE. Find a general solution. For IVPs, solve completely. (a) (b) (c) (d) (e) (f) (g) (h) y 3 y 0 + x3 = 0 y 0 = sec2 y y 0 sin 2πx = πy cos 2πx xy 0 = y 2 + y, (Set xy 0 = x + y, (Set y x = u) y x = u) xy 0 + y = 0, y(4) = 6 y 0 = 1 + 4y 2 , y(1) = 0  xy 0 = y + 3x4 cos2 xy , y(1) = 0, (Set y x = u) 2. Test for Exactness. If exact, then solve. If not, then use an integrating factor. If IVP, then solve completely. (a) (b) (c) (d) (e) sin x cos ydx + cos x sin ydy = 0. e3θ (dr + 3rdθ) = 0. (x2 + y 2 )dx − 2xydy = 0. 3(y + 1)dx = 2xdy, (y + 1)x−4 e2x (2 cos ydx − sin ydy) = 0, y(0) = 0 3. Find the general solution for the linear ODEs. If IVP, then solve completely. (a) (b) (c) (d) (e) (f) y 0 − y = 5.2 y 0 = 2y − 4x y 0 + ky = e−kx y 0 + 2y = 4 cos 2x, y( 41 π) = 3 xy 0 = 2y + x3 ex y 0 + y tan x = e−0.01x cos x, y(0) = 0 1 Modeling 1. Heating and cooling of a building. can be modeled by the ODE Heating and cooling of a building T 0 = k1 (T − Ta ) + k2 (T − Tω ) + P, where T = T (t) is the temperature in the building at time t, Ta the outside temperature, Tω the temperature wanted in the building, and P the rate of increase of T due to machines and people in the building, and k1 and k2 are (negative) constants. Solve this ODE, assuming P =const, Tω =const, and Ta varying sinusoidally over 24 hours, say Ta = A − C cos( 2π 24 t). Discuss the eect of each term of the equation on the solution. 2. are used in physics for accelerating charged particles. Suppose that an alpha particle enters an accelerator and undergoes a constant acceleration that increases the speed of the particle from 103 m/sec to 104 m/sec in 10−3 sec. Find the acceleration a and the distance travelled during that period of 10−3 sec. Linear Particle Accelerators 2 ...
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