## Problem 05 | Substitution Suggested by the Equation

**Problem 05**

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

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**Problem 05**

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

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**Problem 03**

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

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**Problem 02**

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

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**Problem 01**

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

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**Problem 01**

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

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**Problem 02**

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

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**Problem 03**

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

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**Problem 04**

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

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The following are the four exact differentials that occurs frequently.

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

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

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

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

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02 - Solution to Radical Equations

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

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

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

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