First Order Exact DEQ

Conditions

A DEQ in the form:

$$M(x,y)dx+N(x,y)dy=0$$

Where $\frac{\partial f}{\partial x}=M$ and $\frac{\partial f}{\partial y}=N$ have equal mixed partials (condition for reconstructing)

All separable equations are exact but you should probably just solve them as separable

Solving

1. Check exactness: $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ (proving |Clairaut Conditions)
2. Set $\frac{\partial f}{\partial x}=M$ and $\frac{\partial f}{\partial y}=N$
3. Reconstructing from Partials

Integrating Factors

• A method to turn equations that fail the exactness check into exact equations
• Steps
  1. See if either of these only have a dependance on their one variable
     • x: $\frac{M_y^{\prime }-N_x^{\prime }}{N}$
     • y: $\frac{N_x^{\prime }-M_y^{\prime }}{M}$
  2. $\mu =e^{\int \text{above}}$
  3. Multiply $M$ and $N$ by $\mu$ and then solve normally