## Problem 02 | Substitution Suggested by the Equation

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

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

## Separation of Variables

Given the differential equation

$M(x, y)\,dx + N(x, y)\,dy = 0$   ←   Equation (1)

where M and N may be functions of both x and y. If the above equation can be transformed into the form

$f(x)\,dx + f(y)\,dy = 0$   ←   Equation (2)

where f(x) is a function of x alone and f(y) is a function of y alone, equation (1) is called variables separable.