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Line 5:
Line 5: − 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>, since <math>(\frac{\partial \phi}{\partial t}) dt + (\frac{\partial \phi}{\partial y}) dy = 0</math> − + − :<math>\phi(t, y) = \int_0^t M(s, 0) ds + \int_0^y N(t, s) ds</math>+ + + + Line 17:
Line 19: − (That's the condition for "exactness" of the differential form M dt + N dy.) − − − where <math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>.
Exact differential equation (view source)
Revision as of 18:46, August 2, 2010
, 18:46, August 2, 2010no edit summary
:<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>
Before we begin solving it, we must first check that the equation is exact. This means that:
φ is found by integrating M and N:
<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>
To find the solution of this equation, we assume that the solution is φ = constant. We can re-write a different form of this equation by substituting <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>. This yields <math>(\frac{\partial \phi}{\partial t}) dt + (\frac{\partial \phi}{\partial y}) dy = 0</math>, which makes sense.
to find φ, we integrate M with respect to t and N with respect to y. This will give us two different equations. To find φ , we
Go through the example to find φ by integrating, then check that
Go through the example to find φ by integrating, then check that
and that any function φ = some constant, when turned into the corresponding dy/dt, satisfies the original equation. Be sure to emphasize that one must check first that
and that any function φ = some constant, when turned into the corresponding dy/dt, satisfies the original equation. Be sure to emphasize that one must check first that
:<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>
:<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>
To find <math>y</math>, first set <math>M = \frac{\partial \phi}{\partial t}</math> and <math>N = \frac{\partial \phi}{\partial y}</math>. Then manipulate to get <math>M \partial t = \partial \phi</math> and <math>N \partial y = \partial \phi</math>. Integrate both sides, compare the results for <math>\phi</math>, and combine the terms into one equation (for terms that show up in both expressions, only write once in the combined expression.) To solve the expression for <math>y</math>, plug into the quadratic formula.
To find <math>y</math>, first set <math>M = \frac{\partial \phi}{\partial t}</math> and <math>N = \frac{\partial \phi}{\partial y}</math>. Then manipulate to get <math>M \partial t = \partial \phi</math> and <math>N \partial y = \partial \phi</math>. Integrate both sides, compare the results for <math>\phi</math>, and combine the terms into one equation (for terms that show up in both expressions, only write once in the combined expression.) To solve the expression for <math>y</math>, plug into the quadratic formula.