Changes

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>

The solution is &phi; = constant, where &phi; is determined by integrating M and N.

To find the solution of this equation, we assume that the solution is &phi; = constant, where &phi; 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.)