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How to classify ODEs based on their order and whether they are linear or nonlinear examples
There is absolutely no doubt that your instructor will ask you to classify the order and linearity of ODEs during an exam.
We’ve seen it countless times, you can easily get a dozen of these as multiple choice questions in a test, and instructors love them because they’re multiple choice, so they’re very easy for them to mark, and they can stuff a lot of these questions in easily. Once, these made up a quarter of our midterms for an introductory differentials equations course.
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It always breaks our heart when we see students lose easy marks because they don’t understand order or linearity for ODEs, they focused too much on actually knowing how to solve ODEs, but not enough time understanding how to classify them and the concepts behind that!
It sounds really strange, but it’s true – there are actually courses for senior pure or applied mathematics university students that exist, such as Real Analysis, which is entirely focused on using methods to find if abstract math problems are solve-able or not.
So although it does sound off, you can actually get multiple choice questions in a math exam for an ODEs course!
How do classify order and check whether an ODE is linear or nonlinear
To classify order, it’s just the number that’s the highest derivative you can find! So if the highest derivative is second derivative, the ODE is second order!
Sometimes you will see different notations for derivatives, a third derivative might be \(\frac{{{d^3}y}}{{dt}}\) or \(y”’\) (the formatting glitched a bit, that’s a y with three ‘s), so know differences between Leibniz notation or Newton notation.
Linearity is a little trickier, please use our steps:
- You need to find the dependent variable or any of its derivatives, and check if it’s being multiplied by itself in any way, being multiplied with its derivatives, or being found in trigonometric, e’s, logarithms, etc. if so, it’s nonlinear
- Otherwise, as long as the dependent variable is multiplied by a constant number, is multiplied by the independent variable, or doesn’t appear, it’s linear.
- Whatever happens to the independent variable doesn’t affect linearity at all!
Let’s do some examples to show order and linearity of ODEs. We will be extremely thorough and cover many, many cases to ensure the concepts are drilled in your head! If you have any problems, feel free to contact us!
Example problem: \((1 – x)y” – 4xy’ + 5y = cos(x)\)
It’s second-order, linear.
Example problem: \((1 – x)y” – 4xy’ + 5y = cos(y)\)
It’s second-order, nonlinear, because the dependent variable, y, is inside a nonlinear operation, trigonometry.
Example problem: \((1 – x)y”y – 4xy’ + 5y = cos(x)\)
It’s second-order, nonlinear, because the dependent variable, y, is being multiplied by one of its derivatives.
Example problem:\((1 – x)y” – 4xy’ + 5\sqrt y = cos(x)\)
It’s second-order, nonlinear, because the dependent variable, y, is being multiplied by itself, in this case a square root operation which means to the power of 1/2.
Example problem: \((1 – x)y” – 4xy’ + 5y = cos(x) + {e^y}\)
It’s second-order, nonlinear, because the dependent variable, y, is being used in a nonlinear operation, in this case, e.
Example problem: \((1 – x)y” – 4xy’ + 5y = cos(x) + \ln (y)\)
It’s second-order, nonlinear, because the dependent variable, y, is being used in a nonlinear operation, in this case, the natural logarithm, ln.
Example problem: \((1 – x)y” – 4xy’ + 5y = cos(x) + \ln (x) + {e^x} + \sqrt x \)
It’s second-order, linear, remember that no matter what happens to the independent variable, the independent variable won’t affect the linearity of an ODE!
You should be getting the ropes now, let’s spice things up with different looking examples.
Example problem: \(\frac{{{d^2}y}}{{d{x^2}}} = \sqrt {1 + {{\left( {\frac{{dy}}{{dx}}} \right)}^2}} \)
Second-order, nonlinear because y or its derivatives is being multiplied by itself.
Make sure you know nomenclature, see how the second-derivative looks on the left-hand side? No problem with second-derivatives as long as it’s not multiplied by y or another derivative.
Whereas the right-hand side is the first derivative to the power of 2, which is non-linear!
Example: \(x” – \left[ {1 – {{\left( {x’} \right)}^2}} \right] + x = 0\)
Second-order, nonlinear.
Careful here, before y was dependent and x was independent, but here, you need to assume x is dependent and t is independent.
It may sound unusual because you might be used to seeing x used as an independent variable, but it really depends on the context of the problem. x is often used as a dependent variable against the independent variable time, usually x represents displacement, but it can be used to represent a wide variety of natural phenomena.
Also remember in this notation that only the dependent variable can have the prime symbol to show its derivative. Remember before how y was the dependent variable and we saw y’ representing the first derivative of y with respect to the “independent variable”? Well now we see x”, meaning that we have the second derivative of x with respect to the “independent variable”.
Since x, the dependent variable, is multiplied by itself, it’s nonlinear.
Often, if you see only one kind of variable letter in the equation and not two variables, the other variable is likely “t”, but it’s not shown in the equation. And remember that if there’s ever time “t” as a variable, it’s almost always going to be the independent variable. In this case we had the second derivative of x with respect to time t.
Example: \(\frac{{{d^2}u}}{{d{r^2}}} + \frac{{du}}{{dr}} + u = \cos (r + u)\)
Second-order, nonlinear.
The dependent variable is u and it’s inside the cosine term, which is nonlinear.
Again, you may be confused because these variable letters are different than what you are used to, but look into application problems on the ordinary differential equations from heat and mass transfer, often “r” is used as a radius as an independent variable.
There you go, you should be an expert on how to classify your ODEs now!
You won’t get tricked by your instructors, go score some easy marks!