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When you take the FE Exam, without a doubt, you are going to deal with nonhomogeneous differential equations on some level.
The goal right now shouldn’t necessarily be how to solve for the general solution, or what form the particular solution will take.
That will all come in time.
Right now, the main focus should simply be knowing how to identify a non homogeneous equation – fast.
At a fundamental level, you need to know what makes a non homogeneous differential equation different from a homogeneous differential equation.
Once you are solid on this front, you can then worry about the characteristic equation, complementary equation, and everything else that goes into defining the complete solution.
So with all that –
How do you identify NonHomogeneous Differential Equations?
The key to identifying nonhomogeneous differential equations is:
Determining whether or not there are terms present that contain anything other than the dependent variable.
If such terms exist, then we have “forcing functions”, or “source terms”, which indicate that we are indeed dealing with a nonhomogeneous differential equation.
What is an example of a NonHomogeneous Differential Equation?
To illustrate this, say we were given the original differential equation:
- dy/dx + 2y = 3x + 1
The dependent variable, in this case, would be y as we are being asked to solve for the value of y as a function of x, dy/dx.
It’s “dependent” because its value depends on the values of the other variables within the non homogeneous equation.
Now the question we must ask at this point to determine if this is a non homogeneous differential equation or a homogeneous differential equation is this:
Are there terms that include anything other than the dependent variable, y?
In this particular linear equation, there is.
The differential equation has the term 3x + 1 on the right side of the equation which doesn’t include the dependent variable y.
This is a forcing function and confirms that we are dealing with a nonhomogeneous linear differential equation
We can now move forward with our preferred method of solving (undetermined coefficients, variation of parameters, etc) to determine the specific solution.
Constant Coefficients and NonHomogeneous Differential Equations
Another way to quickly identify a nonhomogeneous differential equation is to look for a term that includes a constant non-zero value.
For example, say instead that our original differential equation was written as:
- dy/dx + 2y = 3
We no longer have the 3x term on the right side of the equation.
However, the constant coefficient 3 is still considered to be a term absent of the dependent variable, and because of this, we are again dealing with a nonhomogeneous differential equation.
A constant coefficient or variable term that includes anything other than the dependent variable is considered a forcing function representing external inputs or influences that affect the system being described by the original equation.
So that’s an important note.
We are looking for both constant coefficients and variables to help us identify nonhomogeneous differential equations.
Practice identifying NonHomogeneous Differential Equations
With all of this being established, let’s run through a fundamental set of equations to get in some practice.
Whether we see first order or second order linear equations, the process is the same in determining if we are dealing with a nonhomogeneous equation or a homogeneous equation.
Let’s get into it.
1: dy/dx + 2y = 0
The first step is to define the dependent variable in the given differential equation.
Here we see the term dy/dx, which indicates that we are solving for the derivative of a function y with respect to x – making the dependent variable y.
The second step is to determine if there are any terms in the given differential equation that include variables other than y or non-zero constant coefficients – there are not.
Therefore, we can conclude that the original equation is a homogeneous differential equation because it contains no forcing functions.
2: dy/dx + 2y = 3x + 1
The first step is to define the dependent variable in the given differential equation.
Again, we see the term dy/dx indicating that we are solving for the derivative of a function y with respect to x – the dependent variable is y.
The second step is to determine if there are any terms in the given differential equation that include variables other than y or non-zero constant coefficients – there is 3x + 1.
Therefore, we can conclude that the original equation is a nonhomogeneous differential equation because it contains a forcing function.
3: 3dy/dx – y = sin(x)
What is the dependent variable? y
Are there any terms that include any other variables or non-zero constants? Yes, sin(x)
This is a nonhomogeneous differential equation because it contains a forcing function.
4: y’’ + 4y’ + 4y – 3e^x = 0
What is the dependent variable? y
Are there any terms that include any other variables or non-zero constants? Yes, – 3e^x
This is a nonhomogeneous differential equation because it contains a forcing function.
5: y’’ + 4y’ = -4y
What is the dependent variable? y
Are there any terms that include any other variables or non-zero constants? No
This is a homogeneous differential equation because it does not contain any forcing functions.
6: ds/dt – s^3 = 0
What is the dependent variable? s
Are there any terms that include any other variables or non-zero constants? No
This is a homogeneous differential equation because it does not contain any forcing functions.
7: d^2y/dx^2 + y = x^2
What is the dependent variable? y
Are there any terms that include any other variables or non-zero constants? Yes, x^2
This is a nonhomogeneous differential equation because it contains a forcing function.
8: d^2y/dx^2 = 4dy/dx – 4y
What is the dependent variable? y
Are there any terms that include any other variables or non-zero constants? No
This is a homogeneous differential equation because it does not contain any forcing functions.
9: dy/dx + y^2 = 0
What is the dependent variable? y
Are there any terms that include any other variables or non-zero constants? No
This is a homogeneous differential equation because it does not contain any forcing functions.
10: dy/dx – x^3 + y^2 – z^2 = 0
What is the dependent variable? y
Are there any terms that include any other variables or non-zero constants? Yes, -x^2 and -z^2
This is a nonhomogeneous differential equation because it contains a forcing function.
Why you need to identify NonHomogeneous Differential Equations fast
There are many reasons why students are struggling to pass the FE Exam.
From how they prep and structure their days to the specification they choose to tackle.
But when it comes down to performing on exam day, it doesn’t matter how well you prepared if you aren’t able to manage the time you are given.
The concepts on the FE Exam are wide ranging and require quick, yet deep thinking – unfortunately, for many engineers, deep thinking often leads to over thinking.
If we aren’t able to identify the types of problems we are being presented quickly and efficiently, we are significantly reducing our chances of success on the FE Exam.
This is something we can’t allow.
Without a doubt, you will see nonhomogeneous differential equations on your exam, question is, will you know what you are looking at?
We believe you will, if you remember these two simple questions:
- What is the dependent variable?
- Are there terms that include any other variables or non-zero constants?
If the answer to this second question is yes, then you are looking at a nonhomogeneous differential equation – it’s now time to solve it.
What’s the next step?
If you feel you have wasted time trying to study but haven’t moved the needle any closer to passing the FE Exam.
If you feel you are just guessing on what needs to be done instead of having a clear picture on what it will take.
If you feel passing the FE is far past your capabilities.
Then know this.
Passing the FE Exam shouldn’t be a mystery.
Sadly though, many are doubting that they can get it done, and in turn, are giving up on themselves and their careers.
I don’t want that to be you – if I offered to show you a system that has worked for thousands in your same shoes, would you take it?
