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Unformatted text preview: (c) This little equation is equivalent to one that is linear with x as the dependent variable and y the independent variable(!): This linear equation has μ = exp(y) as an integrating factor. Using the standard recipe, a one-parameter family of solutions is given by . TEST2/MAP2302 Page 4 of 4 ______________________________________________________________________ 7. (10 pts.) The nonzero function f ( x ) = exp( x) is a solution to the homogeneous linear O.D.E. (*) (a) Reduction of order with this solution involves making the substitution into equation (*) and then letting w = v ′ . Do this substitution and obtain in standard form the first order linear homogeneous equation that w must satisfy. (b) Finally, obtain an integrating factor, μ , for the first order linear ODE that w satisfies and then stop. Do not attempt to actually find v. (a) If we have , then , and . Substituting y into (*), and then replacing v ′ using w implies that w must be a solution to after one cleans up the algebra a little. (b) An integrating factor for the homogeneous linear equation that w satisfies is since . _________________________________________________________________ Silly 10 Point Bonus: Suppose that the function , where n is a positive integer, is a solution to the constant coefficient homogeneous linear equation (*) where m > n . What can you say about the coefficients of the ODE, and what can you say about its fundamental set of solutions?? Why??? [Say where your work is, for it won’t fit here.] Appropriate noise may be found on the bottom of Page 2 of 4....
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- Fall '08
- Vector Space