Study Guide for Separable Equation

1. Overview

What are separable equation and how to solve it ?

A differential equation is considered separable if the two variables can be moved to opposite sides of the equation. This facilitates solving a homogenous differential equation, which can be difficult to solve without separation.We are now going to start looking at nonlinear first order differential equations. The first type of nonlinear first order differential equations that we will look at is separable differential equations

A separable differential equation is any differential equation that we can write in the following form.

Note that in order for a differential equation to be separable all the y‘s in the differential equation must be multiplied by the derivative and all the x‘s in the differential equation must be on the other side of the equal sign. Also Consider the equation . This equation can be rearranged to . Any equation that can be manipulated this way is separable. The equation is solved by integrating both sides, resulting in an implicit solution. If an initial condition is provided, you can solve the implicit solution for an explicit solution, and determine the interval of validity, the range of x where the solution is valid. The interval of validity must be continuous and must contain the x-value given in the original condition.

2. Example  

Lets solve a simple question about separable equation.  differential equation: 2(dy)/(dx)=(y(x+1))/(x).                                                                       First of all we have to separate the variables that we have to move all the xs to one side and all the ys to the other. so first step we do is multiply dx on both side so we can cancel the y on the right side so it will give us                              (2dy)=(y(x+1)dx)/(x) and then we divide y on both sides too, so we can cancel the y on the right side. This will give us (2dy)/(y)=(1+1/x)dx  . Now all the y in one side and x is in the other side so we can solve it now by taking the integral on both sides. Therefor

(integral) (2dy)/(y)=integral(1+1/x)dx

So use the answer is 2 ln y= x+ln x+C

3. Video

Here is some videos about separable equation that I copy form KHAN academy if you still confuse about separable equation please watch and I think It will help you a lot.

This is video explain what is separable equation, watch the video for the detail.

In this video, he is explain some example.

Again some example that how to solve the separable equation correctly.

Best of luck on final.