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Line 3:
Line 3: − + − The solution is φ = constant, where φ is determined by integrating M and N.+ Line 16:
Line 16: − − − <math>M(t,y) + N(t,y)y' = 0</math> or <math>M(t,y) dt + N(t,y) dy = 0\,</math>
Exact differential equation (view source)
Revision as of 18:35, August 2, 2010
, 18:35, 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 y' + N = 0 \</math> or <math>M dy + N dt = 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, where φ is determined 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>
:<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>
:<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>
(That's the condition for "exactness" of the differential form M dt + N dy.)
(That's the condition for "exactness" of the differential form M dt + N dy.)