# A04 - y that is x y and use the identities dy dx = 1 dx dy...

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Math 138 – Fall 2011 Assignment 4 Due Friday October 14 (in the drop box by 12 noon) Hand in the following: 1. Consider the complex number z = x + iy where i is the imaginary unit (a constant). Recall that we can rewrite this in polar form as z ( θ ) = r cos θ + ir sin θ . For the following, assume θ and z are the only variables. (a) Compute dz . (b) Based on part a) create a diﬀerential equation dz = f ( z ). That is, ﬁnd the function f ( z ). (c) Solve the DE from part b) by using an initial condition for z (0) according to the deﬁnition of z ( θ ). (d) Deduce Euler’s identity e = cos θ + i sin θ . 2. Solve the IVP y 00 = yy 0 with y (0) = 1 ,y 0 (0) = 1. (Note this is a second order DE but we will convert it to ﬁrst order. During the process you should end up with two arbitrary constants; hence the two initial conditions. Fear not, the following hints will help you along the way.) Hint 1: Assume x can be written as a function of

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Unformatted text preview: y , that is x ( y ), and use the identities: dy dx = 1 dx dy , d 2 y dx 2 =-d 2 x d 2 y 1 ± dx dy ² 3 or, more concisely y = 1 x , y 00 =-x 00 ( x ) 3 to rewrite the DE. Hint 2: Try a substitution u = x (once you’ve manipulated the DE). 3. A tank initially contains 5 kg of salt dissolved in 100 L of water. A solution with a salt concentration of 0.4 kg/L is added at a rate of 5 L/min. The solution is kept mixed and is drained from the tank at the same rate. If y ( t ) is the amount of salt in the tank (in kg) after t minutes, write down the diﬀerential equation satisﬁed by y . Solve this diﬀerential equation to ﬁnd the amount of salt: (a) after t minutes (b) after 30 minutes (c) after a suﬃciently long time 2...
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A04 - y that is x y and use the identities dy dx = 1 dx dy...

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