Ordinary Diff Eq Exam Review Solutions 29

# Ordinary Diff Eq Exam Review Solutions 29 - 1 Solutions 31...

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Unformatted text preview: 1 Solutions 31 6. Since and are homogeneous of degree 2 their quotient is a homogeneous function. Let . Then and the given diﬀerential equation becomes Simplifying and separating variables gives get and so . for it is easy to see that is homogeneous. Let . Clearly . Then . are equilibrium solution. . Integrating gives Separating variables gives and so . Integrating we we get . Since 7. In standard form we get Simplifying gives . Since . Now substitute . The equilibrium solutions imply to get are also solutions. 8. In standard form we get . It is straightforward to see is homogeneous of degree one. So that is a homogeneous diﬀerential equation. Let or . It follows that . Integrating rium solution. Separating variables gives gives then is an equilib- . (To integrate the left hand side use the trig substitution . Now let .) Exponentiating gives . Then , . The case where gives the equilibrium solution. 9. Note that although is part of the general solution it does not satisfy the initial value. Divide both sides by to get . Let . Then . Substituting gives or . An integrating factor is . So . Integrating both sides gives , where we have used integration by parts to compute . Solving for gives . Now substitute ...
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## This note was uploaded on 12/22/2011 for the course MAP 2302 taught by Professor Bell,d during the Fall '08 term at UNF.

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