First-order differential equations -Separable equations

First-order differential equations
The most general first order differential equation can be written as

There is no general formula for the solution to (1) . What we will do instead is look at several special cases and see how to solve those. We will also look at some of the theory behind first order differential equations as well as some applications of first order differential equations. However, in this tutorial we review four of the most commonly-used analytic solution methods for first-order ODES.
1. Separable equations
A first-order differential equation is separable if it can be written in the form

where the function is independent of and is independent of
We can find the general solution of this differential equation by integral



Example 1: Solve the differential equation
Solution:
Rearranging, we have







Example 2: Solve with
Solution: We change to differential form, separate the variables, and integrate





Example 3: Find the general solution of the differential equation
Solution : From the given equation, we have






H.W
Find the general solution of the differential equations


Find the particular solution of the differential equations