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Exact differential equation (view source)
Revision as of 18:38, August 2, 2010
, 18:38, August 2, 2010no edit summary
Suppose you are given an equation of the form:
Suppose you are given an equation of the form:
<math>M(t,y) + N(t,y)y' = 0\,</math> or <math>M(t,y) dt + N(t,y) dy = 0\,</math>
<math>M(t,y) + N(t,y)y' = 0\,</math>
or
<math>M(t,y) dt + N(t,y) dy = 0\,</math>
To find the solution of this equation, we assume that the solution is φ = constant.
This means that <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>
φ is found by integrating M and N:
:<math>\phi(t, y) = \int_0^t M(s, 0) ds + \int_0^y N(t, s) ds</math>
:<math>\phi(t, y) = \int_0^t M(s, 0) ds + \int_0^y N(t, s) ds</math>