## Problem 04 | Exact Equations

**Problem 04**

$(y^2 - 2xy + 6x) \, dx - (x^2 - 2xy + 2) \, dy = 0$

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**Problem 04**

$(y^2 - 2xy + 6x) \, dx - (x^2 - 2xy + 2) \, dy = 0$

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**Problem 03**

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

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**Problem 02**

$(6x + y^2) \, dx + y(2x - 3y) \, dy = 0$

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**Problem 01**

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

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The following are the four exact differentials that occurs frequently.

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

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

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

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

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**.

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