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 eect 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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