# Solve the IVP: 1. (e^x + 2y)dx + (2x - siny)dy = 0, y(0) = 2 pi 2. {dy}/{dx}={e^x}/{1 + e^x},...

## Question:

Solve the IVP:

{eq}1. (e^x + 2y)dx + (2x - siny)dy = 0, y(0) = 2 \pi\\ 2. \frac{dy}{dx}=\frac{e^x}{1 + e^x}, y(0)=4 {/eq}

## Separable Differential Equation:

An equation involving a dependent variable {eq}y {/eq}, an independent variable {eq}x {/eq} and derivative of the dependent variable with respect to an independent variable is known as a differential equation.

A separable differential equation is of the type {eq}\dfrac{dy}{dx} = f \left( x \right) g \left( y \right) {/eq}.

Then, this differential equation can be solved by separating {eq}x \text{ and } y {/eq} variables and then integrating with respect to their derivatives.

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Given the differential equation is

{eq}\left( e^{x}+2y \right)dx+ \left( 2x- \sin y \right)dy = 0 {/eq}

This...