EXACT EQUATIONS

Exact Equations

• DE is in exact form if $(\Psi(x,y))' = 0$, for some $\Psi$, or
  
  
  $$\frac{\partial\Psi}{\partial x}+ \frac{\partial\Psi}{\partial y}\frac{dy}{dx} = 0.$$
  
  

• Test for exact form: is
  
  $$M(x,y) + N(x,y)y' = 0, \mbox{\ with\ } M_y = N_x\ ?$$
  
  

• Then the solution is
  
  $$\Psi(x,y) = \displaystyle\int M(x,y)dx + \displaystyle\int[ N(x,y)-\displaystyle\int M_y(x,y)dx ]dy .$$
  
  

• An integrating factor is sometimes useful:
  given M(x,y) + N(x,y)y' = 0, with $M_y \neq N_x$,
  maybe there is $\mu$, so that $(\mu M)_y = (\mu N)_x$.
  This would mean $M\mu_y - N\mu_x + (M_y-N_x)\mu = 0$.
  E.g. if $\mu= \mu(x)$, then $\mu$ must satsify $\mu' = \frac{M_y-N_x}{N}\mu$
  or if $\mu= \mu(y)$, then $\mu$ must satsify $\mu' = \frac{N_x-M_y}{M}\mu$.