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asdf1
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why is the answer to a differential operator the same as the answer to the original O.D.E. equation?
Actually, I would argue that that doesn't make sense either- an operator is not an equation. An equation may have a solution, but not the operator!dextercioby said:I think he means the "fundamental solution" of the differential operator.
Daniel.
asdf1 said:for example:
(D^2-d-2)y=0
if you solve D, which is D=2,-1
A differential operator is a mathematical operator that operates on a function to produce another function. It is often represented by symbols such as ∂, d/dx, or Δ, and it involves taking derivatives or differences of the function being operated on.
Differential operators are used in many areas of mathematics and science to describe relationships between variables and to solve differential equations. They are especially useful in physics, engineering, and other fields where there are dynamic systems that can change over time.
A derivative is a mathematical operation that gives the rate of change of a function with respect to one of its variables. A differential operator, on the other hand, is a symbol or expression that represents this operation. In other words, a differential operator is a shorthand way of writing a derivative.
Some common types of differential operators include the gradient, divergence, and curl operators in vector calculus, the Laplace operator in differential geometry, and the partial derivative operator in multivariable calculus.
Differential operators are used in a wide range of real-world applications, such as modeling physical systems like fluid flow or heat transfer, analyzing financial markets, and understanding the behavior of biological systems. They are also essential in the development of technologies such as computer graphics, signal processing, and machine learning.