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# A5 - v t for any t> 0(up until the object strikes the...

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Math 128: Assignment 5 due Wednesday, Oct 26 by 4:30 pm Koray Karabina ([email protected]) 1. Use Euler’s method with step size h = 0 . 2 to estimate y (1), where y ( x ) is the solution of the initial value problem: y 0 = y ( x + 1) , y (0) = 1 2. ( Separable Differential Equations ) Solve the differential equations in parts (a)–(e) (a) dP dt = Pt, P (1) = 2 (b) dL dt = kL 2 ln t, L (1) = - 1 ( k is a non-zero constant) (c) dx dt = e x sin t (d) dy dx = 1 - y 2 (e) dy dt = 2 + e y 3. An object of mass m falling near the surface of the earth is retarded by air resistance proportional to its velocity so that, according to Newton’s second law of motion, m dv dt = mg - kv, where v = v ( t ) is the velocity of the object at time t , and g is the acceleration of gravity near the surface of the earth. Assuming that the object falls from rest at time t = 0, that is, v (0) = 0, find the velocity
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Unformatted text preview: v ( t ) for any t > 0 (up until the object strikes the ground). Show v ( t ) approaches a limit as t → ∞ . 4. ( Linear Diﬀerential Equations ) Solve the diﬀerential equations in parts (a)–(b) (a) dy dx-2 y x = x 2 , y (1) = 2 (b) x dy dx-4 y = x 5 e x , y (1) = e 5. (a) A Bernoulli diﬀerential equation is of the form dy dx + P ( x ) y = Q ( x ) y n Observe that if n = 0 or n = 1, the Bernoulli equation is linear. For other values of n , show that the substitution u = y 1-n transforms the Bernoulli equation into the linear equation du dx + (1-n ) P ( x ) u = (1-n ) Q ( x ) (b) Use the method in part (a) to solve the diﬀerential equation dy dx + 2 x y = y 3 x 2...
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