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Exact differential equation

2 bytes removed, 18:55, 2 August 2010
To find the solution of this equation, we assume that the solution is &phi; = constant. We assume that <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>. (If we substitute M and N back into (1), it yields <math>(\frac{\partial \phi}{\partial t}) dt + (\frac{\partial \phi}{\partial y}) dy = 0</math>, which makes sense.)
To find <math>y/,</math>, manipulate the substitutions of M and N to get <math>M \partial t = \partial \phi</math> and <math>N \partial y = \partial \phi</math>. Integrate both sides. To get the main function &phi; write the sum of each term found in each equation. For terms that appear in both equations, only write them once.
To find <math>y</math>, manipulate the substitutions of M and N to get <math>M \partial t = \partial \phi</math> and <math>N \partial y = \partial \phi</math>. Integrate both sides. To get the main function &phi; write the sum of each term found in each equation. For terms that appear in both equations, only write them once.  To solve the expression for <math>y/,</math>, use the quadratic formula.
[[Category:Calculus]]
[[Category:Differential Equations]]