DE Tutorial - Part I: First Order Equations and Homogeneous Second Order Equations

Once in a while, differential equation questions pop up, so I'm going to point out the various techniques on how to solve them:

** 1. Direct Integration**:

If you have a differential equation in the form , we can use direct integration to solve the DE.

__Example 1__:

*Solve*

To solve, simply integrate both sides of the equation:

Note that we can combine the two constants into a new constant C.

Also, we may encounter differential equations with given conditions. These types of differential equations are called **initial value problems (IVP)**. When solving a DE without conditions, we always find the **General Solution** to the DE. When an initial condition is applied, then we are finding a **Particular Solution**. Let's go through a quick example.

__Example 2__:

*Solve *

Directly integrate the DE:

We need to apply integration by parts to the integral on the right side:

let and Therefore, and

Now apply the initial condition

Thus, our particular solution is

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**2. Separation of Variables**

Another technique in solving differential equations is separation of variables. As the name suggests, we "separate" one variable from another in order to find a solution. Some of these are very straight-forward, whereas some of the DE's require some thought. I will go through an easier example, and then a harder one.

__Example 3__:

*Solve*

First, factor out a and then separate the variables:

Integrate both sides and solve for y.

Apply the initial condition

__Example 4__:

*Solve*

Factor the right hand side of the equation:

Separate the variables and integrate:

Solve for y:

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I will post more later on today...after I sleep (Sleepy)