Differential Equations: Orders Explained

In summary, the order in differential equations refers to the highest order of the derivatives, not the actual functions. A function like y + yy' would be considered first order, while y + y'' would be second order. However, a function like y^4 + 3yy' would still be considered first order. It is preferable to use the term "degree of the ODE" instead of "order of the function or its derivative". If the degree is greater than 1, the ODE is considered nonlinear.
  • #1
tandoorichicken
245
0
Okay! Just making sure I have the concept of orders in differential equations right. So the order refers to the highest order of the derivatives, not the actual functions right?

So a function like y + yy' = ? would be first order, y + y'' would be second order, but something like y^4 +3yy' would still be first order right?
 
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  • #2
Yep, you've got it.
 
  • #3
The "order of the functions" (the power that "y" or its derivatives have in the ODE) gives the nonlinear character...

Daniel.
 
  • #4
I guess it's preferable to talk about the "degree of the ODE" instead of the "order of the function or its derivative". If degree > 1, the ODE is nonlinear.
 
  • #5
dextercioby said:
The "order of the functions" (the power that "y" or its derivatives have in the ODE) gives the nonlinear character...
That's "degree". Though the original post should have referred to "order of the diffential equation", not "order of the function".
 

1. What is the difference between first, second, and higher order differential equations?

First order differential equations involve only the first derivative of the unknown function, while second order differential equations involve the first and second derivatives. Higher order differential equations involve derivatives of even higher orders.

2. How do you determine the order of a differential equation?

The order of a differential equation is determined by the highest derivative present in the equation. For example, if the equation contains a third derivative, it is a third order differential equation.

3. What is the general solution to a differential equation?

The general solution to a differential equation is an equation that contains all possible solutions to the given equation. It usually contains a constant or a set of constants, which can be solved for using initial conditions.

4. Can a differential equation have multiple solutions?

Yes, a differential equation can have multiple solutions. This is because there are often many different functions that can satisfy the same differential equation. However, the general solution will contain all of these possible solutions.

5. How are differential equations used in real-world applications?

Differential equations are used in a wide range of scientific fields, including physics, engineering, biology, and economics. They are used to model and analyze various phenomena such as motion, growth, and decay, and to make predictions about the behavior of complex systems.

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