\frac{dy}{dx} = \frac{(2x - 2xy)}{(x^2 + 2y)} Find all points on the curve where x = 2. Show...


{eq}\frac{dy}{dx} = \frac{(2x - 2xy)}{(x^2 + 2y)} {/eq}

Find all points on the curve where x = 2. Show there is a horizontal tangent to the curve at one of those points.

Exact Differential Equation:

An equation with derivative term is known as differential equation.

Equation of form M(x,y)dx+N(x,y)dy=0 is known as exact differential equation when it satisfies the condition {eq}\displaystyle \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} {/eq},

Otherwise for non exact differential equation find integrating factor {eq}\displaystyle \:I.F=e^{\int \mu \left(y\right)dy\:}\\ \displaystyle where,\mu \left(y\right)=\frac{\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}}{M} {/eq} and multiply it with original equation to find solution.

Solution will be

{eq}\displaystyle \int _{y\:consatnt}M\:dx+\int _{No\:x\:term}N\:dy=C\:\: {/eq}

Answer and Explanation:

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Given ODE is

{eq}\displaystyle \frac{dy}{dx}\:=\:\frac{\left(2x\:-\:2xy\right)}{\left(x^2\:+\:2y\right)}\\ \displaystyle...

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Learn more about this topic:

First-Order Linear Differential Equations


Chapter 16 / Lesson 3

In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.

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