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HMWK 1 Solution

# HMWK 1 Solution - CME 102 Winter 2009 Ordinary Dierential...

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CME 102, Winter 2009 Ordinary Differential Equations for Engineers Prof. Eric Darve Homework 1 – Solutions 1. Kreyszig: (a) exercise 6 page 8 This is a second order differential equation. We take twice the derivative of y to get y 00 = - 2 cos( πx ) - 2 sin( πx ) = - π 2 [ a cos( πx ) + b sin( πx )] = - π 2 y. We can thus conclude that y 00 + π 2 y = 0. (b) exercise 16 page 8 Substitution of y l = cx - c 2 into the ODE gives ( y 0 l ) 2 - xy 0 l + y l = c 2 - xc + ( xc - c 2 ) = 0 thus proving that y l = cx - c 2 is the general solution. Similarly, for y p = 1 4 x 2 , we get that ( y 0 p ) 2 - xy 0 p + y p = 2 4 x 2 - x 2 4 x + 1 4 x 2 = 1 4 - 1 2 + 1 4 x 2 = 0 . In figure 6, we can see that the parabola represents the function y p . Such parabola is the envelope of the set of general solutions, the lines y l (recall that c is an arbitrary constant). For a given c , the corresponding line y l is tangent to the parabola y p at exactly one point. (c) exercise 20 page 8 From the physical information, we have that the pressure satisfies an ODE of the form y 0 = ky . The general solution to this ODE is y = y 0 exp( kx ), where y 0 is the pressure at sea level x = 0. Now we also know that y (18000) = 1 2 y 0 = y 0 exp( k. 18000) .

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HMWK 1 Solution - CME 102 Winter 2009 Ordinary Dierential...

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