Second order homogeneous ODE with real, different roots. Today it’s all about the solution of second order homogeneous ODE with real, different roots. Let’s start then.

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### Second order homogeneous ODE with real, different roots

If you’re looking for more in second-order differential equations, do check in:

**Second-order homogeneous ODE with real and equal roots**

**Second-order homogeneous ODE with complex roots**

**What is a particular integral in second-order ODE**

**Exponential functions as particular integrals**

**Trigonometric functions as particular integrals**

Now it’s a known thing that the standard form of any second order homogeneous ODE is

Therefore the auxiliary equation for this ODE is

(1)

Next, I’ll solve the equation (1) to get the values of .

So there are three options for the roots of this equation.

- First option: the roots of the auxiliary or the characteristic equation are real and different.
- Second option: the roots of the auxiliary or the characteristic equation are real and equal.
- Third option: the roots of the auxiliary or the characteristic equation are complex conjugate.

In this post, I’ll work with the first option only.

So, let me choose the solutions of this equation as

Here both and are real numbers.

Therefore the general solution of the differential equation will be

Here both and are constants.

Now I’ll solve some examples.

#### Examples of second order homogeneous ODE with real, different roots

Disclaimer: None of these examples is mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here are the examples.

##### Example 1

According to Kreyszig (2005)*“Find the general solution of the following: ”

##### Solution

Here the given second-order differential equation is

Therefore the auxiliary equation (characteristic equation) will be

Now I’ll solve this equation to get the values of .

So I can also rewrite this equation as

This means

So I can say or .

That shows the given differential equation has two real different roots.

Therefore the general solution of the given differential equation is

This is the solution to the given equation.

Next, I’ll show you another example.

##### Example 2

According to Kreyszig (2005)*“Find the general solution of the following: ”

##### Solution

Here the given second-order differential equation is

Therefore the auxiliary equation (characteristic equation) will be

Now I’ll solve this equation to get the values of .

So I can also rewrite this equation as

Now I can divide this equation throuth out by 2 to get

Next, I’ll factorise it to get

This means

So I can say or .

That shows the given differential equation has two real different roots.

Therefore the general solution of the given differential equation is

Hence this is the solution to the given equation.

Now I’ll give another example.

##### Example 3

According to Kreyszig (2005)*“Find the general solution of the following: ”

##### Solution

Here the given second-order differential equation is

Therefore the auxiliary equation (characteristic equation) will be

Now I’ll solve this equation to get the values of .

So I can also rewrite this equation as

This means

So I can say or .

That shows the given differential equation has two real different roots.

Therefore the general solution of the given differential equation is

Hence I can conclude that this is the solution to the given equation.

Dear friends, this is the end of today’s post. Thank you very much for reading this. Please let me know how you feel about it. Soon I will be back again with a new post. Till then, bye, bye!!

###### Kreyszig, E. (2005): Advanced Engineering Mathematics: International Edition, John Wiley & Sons, 9th Edition, 29th December 2005, Chapter 2, Second-order linear ODEs, p. 59, Problem set 2.2, Q. 1(Example 1), Q. 2 (Example 2), Q. 3 (Example 5).

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