03 Inexact DE

Last Updated: 2 years ago2023-01-25 ; Contributors: AhmedThahir

Consider a 1^st order inexact DE

$$M(x, y) dx + N(x, y) dy = 0, \quad (M_y \ne N_x)$$

Steps

1. Find $M_y - N_x$
2. 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	$e^{\int h(y) \cdot dy}$	$e^{\int g(x) \cdot dx}$

1. Multiply both sides of equation: Inexact DE $\times$ IF $\to$ Exact DE
2. 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

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