## Problem 05 | Substitution Suggested by the Equation

Problem 05
$dy/dx = \sin (x + y)$

## Problem 03 | Substitution Suggested by the Equation

Problem 03
$dy/dx = (9x + 4y + 1)^2$

## Problem 02 | Substitution Suggested by the Equation

Problem 02
$\sin y(x + \sin y)~dx + 2x^2 \cos y~dy = 0$

## Problem 01 | Substitution Suggested by the Equation

Problem 01
$(3x - 2y + 1)~dx + (3x - 2y + 3)~dy = 0$

## Problem 01 | Equations with Homogeneous Coefficients

Problem 01
$3(3x^2 + y^2) \, dx - 2xy \, dy = 0$

## Problem 02 | Equations with Homogeneous Coefficients

Problem 02
$(x - 2y) \, dx + (2x + y) \, dy = 0$

## Problem 03 | Equations with Homogeneous Coefficients

Problem 03
$2(2x^2 + y^2) \, dx - xy \, dy = 0$

## Problem 04 | Equations with Homogeneous Coefficients

Problem 04
$xy \, dx - (x^2 + 3y^2) \, dy = 0$

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

## 02 - Solution to Radical Equations

Determine the value of $x$ from the following equations:

1. $\sqrt{(4 - x^2)^3} + 3x^2\sqrt{4 - x^2} = 0$

2. $\dfrac{1}{3x - 2} - \dfrac{8}{\sqrt{3x - 2}} = 9$