what is a differential eqn?
It is an equation connecting a function with the independent variable, and its
own derivative (or derivatives) so a first order Ordinary Differential Equation
(ODE) is something like:
f(x, y, y')=0
a second order one is of the form:
f(x, y, y', y'')=0,
and so on for higher order.
A Partial Differential Equation (PDE) is similar but there are multiple
independent variables and also uses partial derivatives.
Some examples of ODEs:
1:
y'' = - 3 y,
or in standard form:
y'' + 3 y = 0.
2:
y'' + x y' + x^2 y = sin(3 x),
or in standard form:
y'' + x y' + x^2 y - sin(3 x) = 0.
RonL
There are two types of equations.
(Abusing terminology)
An algebraic equation asks to find all number that satisfies the equation.
For example,
x^5=1
A functional equation asks to find all functions that satisfies the equation.
For example,
f(x+y)=f(x)f(y) for all numbers x,y
Note we are not solving for a number but a function that has that property.
(If you are interesting all exponential functions make that true.)
The problem with functional equation is that there is almost no developed way to solve them. Thus,
f(x+y)=f(x)f(y)
Might have other solutions rather than exponentials (I do not know).
So a differencial equation is a type of functional equation. Differencial equations are much easier to solve because they are "well-behaved" meaning a function behaves normal rather than going all over the place.
It gets its name because the derivative appears there.
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And you blame me for being to complicated.Originally Posted by CaptainBlank
An Ordinary Differential Equation (ODE) is an equation relating the derivative (and higher order derivatives for higher order ODEs) of a function to the values of the function and the independent variable.
For example:
dy/dx + 3 x y = f(x),
for some function of f of x.
RonL
Hello, Akanksha gupta!
I must assume you know what a derivative is . . .
What is a differential equation?
Here's the baby-talk approach I've used in my classes . . .
In algebra, we are asked to solve: .2x - 4 .= .-3
We want a number x so that:
. . if we double x and subtract 4, we get -3.
And there are algebraic procedures for finding the solution (x = ½).
In Differential Equations, we can be given something like:
. . . .dy
. . 2 --- - 4y .= .-3
. . . .dx
We want a function y = f(x) so that:
. . if we double the derivative of y and subtract 4 times y, we get -3.
And there are procedures for solving this differential equation.
Solution: . y .= .Ce^{2x} + ¾ .for any constant C.
Is that not what I said?
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Is there such a thing as a functional-function equation?
For example,
f(x+y)=f(x)+f(y) where f:R--> R
But are there cases where we have,
L(x+y)=L(x)+L(y) where L:R(x)--->R(x)
Where R(x) represents the set of functions f:R-->R
The reason why I am asking this is because the Laplace Transform appears to do that. It transform a function into a function.
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This is to the user who asked the question. If you do not know what a derivative is then you cannot understand well what a differencial equation is.
To add, differencial equations are useless to mathemations. Only for science and physics they are useful. Funny that mathemations developed them but never use them.