Solutions of Differential Equations

rayn123x
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When it is stated that a certain formula is solution to a differential equation, what does that mean in the physical world. What is the significance of a certain formula being a solution to a differential equation?
 
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The value f subscript x gives you the slope of the line tangent to f in the x direction.

EDIT: I think. I learned this like 2 weeks ago.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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