## Problem 02 | Determination of Integrating Factor

Problem 02
$2y(x^2 - y + x)~dx + (x^2 - 2y)~dy = 0$

## Problem 01 | Determination of Integrating Factor

Problem 01
$(x^2 + y^2 + 1)~dx + x(x - 2y)~dy = 0$

## Problem 01 to 05 Perform the indicated operation of the given functions

Problem 1
If f(x) = x2 - x + 3; find f(0), f(2), f(-4), f(-2x).

Problem 2
If f(x) = 7 - 2x + x2, find f(0), f(3), f(-2), f(-y).

Problem 3
If F(y) = y(y - 3)2; find F(c), F(0), F(3), F(-1), F(x + 3).

Problem 4
If F(b) = (b - b2) / (1 + b2), find F(0), F(1), F(1/2), F(tan x).

Problem 5
If g(x) = 4x4 - 3x2 + 2x - 2, find g(2), g(-2), g(1/2), g(-x)

## Integrating Factors Found by Inspection

The following are the four exact differentials that occurs frequently.

1. $d(xy) = x \, dy + y \, dx$

2. $d\left( \dfrac{x}{y} \right) = \dfrac{y \, dx - x \, dy}{y^2}$

3. $d\left( \dfrac{y}{x} \right) = \dfrac{x \, dy - y \, dx}{x^2}$

4. $d\left( \arctan \dfrac{y}{x} \right) = \dfrac{x \, dy - y \, dx}{x^2 + y^2}$
5.

6. $d\left( \arctan \dfrac{x}{y} \right) = \dfrac{y \, dx - x \, dy}{x^2 + y^2}$