Ordinary Diff Eq Exam Review Solutions 33

# Ordinary Diff Eq Exam Review Solutions 33 - 1 Solutions...

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Unformatted text preview: 1 Solutions equilibrium solutions. Separating variable gives So integrating gives solution we get 35 . Now . Substituting, we get the implicit . For the equilibrium solution . 23. This is the same as Exercise 16 where the Bernoulli equation technique there used the substitution . Here use the given substitution to get . Substituting we get and in standard form . Clearly, is an equilibrium solution. Separating variables gives for and integrating gives we get , where and solving for gives the equilibrium solutions , . Since gives . . Solving we get . The case 24. If then . Multiply the given diﬀerential equation by to get . Substituting we get and in standard form . An integrating factor is so . Integrating by parts twice leads to and hence . Solving for gives , . 25. If then . Divide the given diﬀerential equation by . Then and substitution gives so 26. Let . An integrating factor is . Integration (by parts) gives . Finally, solving for we get . Then so , and so . . Substituting gives . Multiply both sides by and put in standard form to get An integrating factor is and so We thus get so . Since , , . Integrating we get and this requires . . . Section 1.6 1. This can be written in the form where and . Since , the equation is exact (see Equation (3.2.2)), and the general solution is given implicitly ...
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