2.2.2 Homogeneous Equations

2.2.2 Homogeneous Equations - GE 207K Summary: Homogeneous...

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GE 207K Homogeneous Equations September 15, 2011 Summary: A diferential equation which can be written in the Form y ° = g ° y x ± . In other words, the diferential equation can be written such that anywhere the dependant variable appears, it appears as Fraction y x . These equations can be turned into separable equations by perForming a substitution. These homogenous equations are diferent than the class oF diferential equations which we classi±ed as homogenous on the ±rst day. The Following steps can be taken to ±nd the general solution to the diferential equation: Steps: 1. Write the given diferential equation in the right Form. y ° = g ° y x ± (1) 2. Eliminate y From the diferential equation entirely by rewriting the diferential equation in terms oF x and (a newly introduced variable) u . Do this by perForming the Following substitutions: (a) PerForm the substitution in the diferential equation u = y x . (2) (b) Replace y ° with the Following expression y = ux, y ° = u ° x + u (chain rule) . (3) 3. IF you’ve done everything right, the diferential equation should be a separable equation at this point (in terms oF u and x only). Solve by separating variables and integrating both sides oF the resulting equation.
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This note was uploaded on 02/03/2012 for the course M 427K taught by Professor Fonken during the Fall '08 term at University of Texas at Austin.

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2.2.2 Homogeneous Equations - GE 207K Summary: Homogeneous...

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