Linear vs nonlinear diff equation II

In summary, for the linear differential equation, a_n and b_n can be either functions of x or constants, but not both. This is because they are using "or" in the exclusive sense. A special case of a "function of x" is f(x) = constant, which can be thought of as a trivial solution. A trivial solution can also mean "not very interesting". The crucial point in "linear" equations is that the coefficients, a_i, cannot be functions of y or any of its derivatives.
  • #1
Calpalned
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For the linear differential equation, it says that a_n and b_n are constants or functions of x. This implies that a function of x = f(x) = constant. How can a number be a constant if it is a function? I think I'm misunderstanding something.
 

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  • #2
They mean that ## a_n ## and ## b_n ## can be either functions of ## x ## or constants, but not both. In other words, they are using "or" in the exclusive sense, not in the inclusive sense. We could also think of ## f(x) = ## constant as a special case of a "function of ##x##", albeit a trivial one. Think of a machine that spits out the same thing, no matter what you feed into it.
 
  • #3
Geofleur said:
They mean that ## a_n ## and ## b_n ## can be either functions of ## x ## or constants, but not both. In other words, they are using "or" in the exclusive sense, not in the inclusive sense. We could also think of ## f(x) = ## constant as a special case of a "function of ##x##", albeit a trivial one. Think of a machine that spits out the same thing, no matter what you feed into it.
If I am not mistaken, a trivial solution is one where the solution is zero?
 
  • #4
Well, trivial can also mean "not very interesting".
 
  • #5
Calpalned said:
If I am not mistaken, a trivial solution is one where the solution is zero?
Yes, the trivial solution is the zero solution.
 
  • #6
The crucial point in "linear" equations is that the coefficients, [itex]a_i[/itex], cannot be functions of y or any of its derivatives.
 

Related to Linear vs nonlinear diff equation II

1. What is the difference between linear and nonlinear differential equations?

Linear differential equations can be expressed in the form of a linear combination of the dependent variable and its derivatives, while nonlinear differential equations cannot be expressed in this form. In other words, linear equations have a constant rate of change, while nonlinear equations have a variable rate of change.

2. How do you solve a linear differential equation?

To solve a linear differential equation, you can use various methods such as separation of variables, integrating factors, and substitution. These methods involve manipulating the equation to isolate the dependent variable and then integrating both sides to find the solution.

3. How do nonlinear differential equations differ from linear ones in terms of solution methods?

Nonlinear differential equations are generally more difficult to solve compared to linear ones. While linear equations have well-defined solution methods, nonlinear equations often require numerical methods or approximations to find a solution.

4. Can a nonlinear differential equation have a linear solution?

No, a nonlinear differential equation cannot have a linear solution. This is because the defining characteristic of a nonlinear equation is that it cannot be expressed in the form of a linear combination of the dependent variable and its derivatives.

5. What is the significance of linear vs nonlinear differential equations in science?

Linear and nonlinear differential equations are essential tools in many scientific fields, including physics, engineering, and biology. Linear equations are often used to model simple systems with constant rates of change, while nonlinear equations are used for more complex systems with varying rates of change. Understanding the differences between linear and nonlinear equations is crucial in accurately describing and predicting real-world phenomena.

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