Coefficients derivative

In summary: Irregular singular points cannot be handled so simply.In summary, we typically write differential equations in the form y''(x)+a(x)y'(x)+b(x)=f(x) because it is standard notation and easier to work with. We do not include coefficients such as m(x) unless necessary, as it can complicate the solution process. If m(x) is equal to 0 for some x, it creates a singular point and can drastically change the behavior of the solution set.
  • #1
matematikuvol
192
0
Why we always write equation in form
[tex]y''(x)+a(x)y'(x)+b(x)=f(x)[/tex]

Why we never write:
[tex]m(x)y''(x)+a(x)y'(x)+b(x)=f(x)[/tex]
Why we never write coefficient ##m(x)## for example?
 
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  • #2
matematikuvol said:
Why we always write equation in form
[tex]y''(x)+a(x)y'(x)+b(x)=f(x)[/tex]

Why we never write:
[tex]m(x)y''(x)+a(x)y'(x)+b(x)=f(x)[/tex]
Why we never write coefficient ##m(x)## for example?

Because usually the first thing to do is divide by the coefficient of [itex]y''[/itex].
 
  • #3
But what if for some ##x##, ##m(x)=0##.
 
  • #4
matematikuvol said:
But what if for some ##x##, ##m(x)=0##.

My guess, the behavior of the solution set changes drastically wherever m(x)=0.
 
  • #5
I'm not sure how to answer your question, "Why we never write coefficient m(x) for example?" because we often do! I suspect you simply have not yet gone far enough in differential equations to see such equations.

Of course, if m(x) is never 0, we can simplify by dividing by it. If m(x)= 0 for some x, that x becomes a "singular point" for the equation- either a "regular singular point" or an "irregular singular point". Regular singular points can be handled in a similar way to "Euler type" or "equi-potential equations, [itex]ax^2y''+ bxy'+ cy= f(x)[/itex] where each coefficient has x to the same degree as the order of the derivative. Such equations are typically approach late in a first semester differential equations class.
 

1. What is a coefficient derivative?

A coefficient derivative is a mathematical concept used to determine how a function changes with respect to its independent variable. It is the slope of the tangent line at a specific point on a graph and represents the rate of change of the function at that point.

2. How is a coefficient derivative calculated?

A coefficient derivative is calculated using the power rule, which states that the derivative of a term with a variable raised to a power is equal to the coefficient multiplied by the power, with the power reduced by one. This rule is applied to each term in the function separately, and the results are added together to find the overall derivative.

3. What is the significance of a coefficient derivative?

The coefficient derivative is significant because it allows us to analyze how a function is changing at a specific point. It can be used to find maximum and minimum values, identify critical points, and understand the behavior of a function. It is also essential in optimization problems and is a crucial concept in calculus.

4. How does a coefficient derivative relate to the slope of a line?

The coefficient derivative and the slope of a line are closely related. The coefficient derivative at a specific point is equal to the slope of the tangent line at that point. This means that the coefficient derivative represents the rate of change of the function, similar to how the slope of a line represents the rate of change between two points on a line.

5. Can a coefficient derivative be negative?

Yes, a coefficient derivative can be negative. A negative coefficient derivative indicates that the function is decreasing at that point, while a positive coefficient derivative indicates that the function is increasing. However, the value of a coefficient derivative only tells us about the slope of the tangent line at that point and does not provide information about the overall behavior of the function.

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