Linear vs nonlinear diff equation II

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In linear differential equations, coefficients a_n and b_n can either be constants or functions of x, but not both simultaneously. This distinction clarifies that when a function is constant, it represents a special case of a function of x, akin to a machine producing the same output regardless of input. The term "trivial solution" refers to the zero solution, which is considered less interesting in this context. A key aspect of linear equations is that the coefficients cannot depend on y or its derivatives. Understanding these definitions is essential for grasping the nature of linear versus nonlinear differential equations.
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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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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.
 
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?
 
Well, trivial can also mean "not very interesting".
 
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.
 
The crucial point in "linear" equations is that the coefficients, a_i, cannot be functions of y or any of its derivatives.
 

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