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Differential Eqs 1[1]

Differential Eqs 1[1] - 29 s k dt d θ-= where is the...

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205FOE102 Further Mathematics 1 Differential Equations 1 Direct Integration 1. Find the general solutions of the differential equations. (a) 1 2 - = x dx dy (b) x x dx dy - = sin (c) 2 1 x e dx dy x - = Separation of Variables 2. Find the general solutions of the differential equations. (a) ( 29 ( 29 1 1 - - = y x dx dy (b) y dx dy x y 4 2 = - (c) xy y dx dy x 3 - = 3. Solve the following boundary value problems. (a) 0 2 2 = + dx dy y x , y = 3 when x = 0 (b) 0 cos 1 = + dx dy y e x , ( 29 3 0 π = y (c) x xy dx dy y 4 2 + = , ( 29 0 0 = y 4. Solve the following initial value problems. (a) 2 cos z t dt dz = , ( 29 1 0 = z (b) u dt du t 4 2 = , u = 2 when t = 1 (c) ( 29 ( 29 1 1 + = - v v dt dv t t , ( 29 2 2 = v
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5. Newton’s Law of Cooling states that the rate of decrease of temperature of a body is directly proportional to the difference in temperature between the body and the surroundings. This is modelled by the differential equation
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Unformatted text preview: ( 29 s k dt d θ--= where is the temperature at time t s is the constant temperature of the surroundings k is a constant Solve this equation given that the initial temperature was = . Sketch the variation of temperature with time. 6. A chemical reaction is governed by the differential equation ( 29 2 3 x K dt dx-= where x is the concentration of chemical at time t and K is the reaction rate constant. Solve the differential equation given that the concentration is zero initially. At time t = 5, the concentration is found to be x = 1. Calculate the reaction rate constant, K . What is the final value of the concentration?...
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