## Problem 05 | Substitution Suggested by the Equation

**Problem 05**

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

**Problem 05**

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

**Problem 03**

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

**Problem 02**

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

**Problem 01**

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

The following are the four exact differentials that occurs frequently.

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

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

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

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