Order of differential equation

In summary, the conversation discusses finding the order of the differential equation for a family of parabolas with a fixed directrix. The number of constants in the general formula is equal to the order of the equation, and differentiating once can eliminate one constant. The general equation involves two parameters, the focus and directrix, and the distance between them can be used to determine the equation of the parabola.
  • #1
lizzie
25
0
Find the order of the differential equation of the following curve:
Family of parabolos having fixed directrix.

Thanks for any help.
 
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  • #2
What does the general equation of such a family look like? How many constants are there in that general formula. Generally speaking you need to differentiate once to eliminate one constant. The number of constants is the order of the equation.
 
  • #3
I am unable to find the general equation. Please give me some hint.
 
  • #4
lizzie said:
I am unable to find the general equation. Please give me some hint.

Hi lizzie! :smile:

(how many dimensions is this in? and is the position of the focus constrained in any way?)

Hint: for a fixed focus and directrix, the parabola is all points whose distance from the focus equals its (perpendicular) distance from the directrix. :smile:
 
  • #5
The vertex of a parabola is exactly half way between the focus and the directrix so a parabola having focus (0, f) and directrix y= -d has vertex at (0, (f-d)/2). The focal distance (distance from the vertex to the focus) is then f- (f-d)/2= (f+d)/2. The equation of such a parabola is y= 2(f+d)x2+ (f-d)/2. How many parameters (constants) does that involve?
 

FAQ: Order of differential equation

What is an order of a differential equation?

The order of a differential equation refers to the highest derivative present in the equation. For example, a first-order differential equation has only first derivatives, while a second-order differential equation has second derivatives.

How is the order of a differential equation determined?

The order of a differential equation is determined by examining the highest derivative present in the equation. It is important to note that the order of a differential equation can change if the equation is manipulated or rewritten in a different form.

What is the significance of the order of a differential equation?

The order of a differential equation can provide information about the complexity of the equation and the number of initial conditions needed to solve it. Higher order differential equations typically require more initial conditions and may be more difficult to solve.

Can a differential equation have a fractional or negative order?

No, a differential equation must have a positive integer order. Fractional or negative orders do not make physical or mathematical sense and would result in an invalid equation.

How does the order of a differential equation affect its solution?

The order of a differential equation can affect the difficulty of solving it and the type of solution that can be obtained. Higher order differential equations may have more complex solutions and may require more advanced mathematical techniques to solve.

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