03 Inexact DE
Consider a 1st order inexact DE
\[ M(x, y) dx + N(x, y) dy = 0, \quad (M_y \ne N_x) \]
Steps¶
- Find \(M_y - N_x\)
- You will get one of the following cases the simplification will give in terms of a single variable
| Case 1 | Case 2 | |
|---|---|---|
| \(\dfrac{M_y - N_x}{\color{orange}-M} = h(y)\) | \(\dfrac{M_y - N_x}{\color{orange}N} = g(x)\) | |
| IF | \(e^{\int h(y) \cdot dy}\) | \(e^{\int g(x) \cdot dx}\) |
- Multiply both sides of equation: Inexact DE \(\times\) IF \(\to\) Exact DE
- Then, use Exact DE method
Shortcut¶
- Try to get everything in terms of simple integrals like \(dx, dy, d(xy),d(x+y)\).
- Then use exact DE formulae
This way we can avoid the IF step