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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 di ff erential equation which can be written in the form y = g y x . In other words, the di ff erential 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 di ff erent than the class of di ff erential equations which we classified as homogenous on the first day. The following steps can be taken to find the general solution to the di ff erential equation: Steps: 1. Write the given di ff erential equation in the right form. y = g y x (1) 2. Eliminate y from the di ff erential equation entirely by rewriting the di ff erential equation in terms of x and (a newly introduced variable) u . Do this by performing the following substitutions: (a) Perform the substitution in the di ff erential 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 di ff erential 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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