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Like with finding solutions of separable ones.. it's just integrating both sides. And with finding other solutions with exact and linear equations, there is always integration.

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- Thread starter rakeru
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- #1

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Like with finding solutions of separable ones.. it's just integrating both sides. And with finding other solutions with exact and linear equations, there is always integration.

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I am talking about first order ones.. I don't know how to solve second order ones.

- #3

pasmith

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f' : x \mapsto \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}[/tex]

A differential equation is an equation in which derivatives of a function appear.

Thus, given [itex]g[/itex], [itex]f'(x) = g'(x)[/itex] is a differential equation with solution [itex]f(x) = g(x) + C[/itex].

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HallsofIvy

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If you have ##y' =f(x)## then ##y = F(x)+C## where ##F## is an antiderivative of ##f## and ##C## is an arbitrary constant. So you get the original function by antidifferentiating (integrating) to within a constant. For a simple DE in that form, you do solve it by integrating, but I wouldn't say it is just another way of integrating. You use integrating to solve it. But differential equations can be more general so that you can't solve them in practice by integrating. What I mean by that is, for example, there is no general method to express the solution of a general DE like ##y'=f(x,y)## with integrals. Even so, I would agree with the statement that in some sense, "there is always integration" underlying the problem. Not sure how meaningful a vague statement like that is though.

Like with finding solutions of separable ones.. it's just integrating both sides. And with finding other solutions with exact and linear equations, there is always integration.

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Simon Bridge

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No.Is the derivative of a function a differential equation?

A differential equation must

i.e. the derivative of the function x

But y'=2x is a differential equation. The LHS is just some notation that tells you that the RHS is the derivative of y.

To be a differential equation it has to include the dervativeI guess it would be because it involves a derivative, right?

the derivative (wrt x) of y

Lets see - the derivative (wrt x) of yWould the solution to the equation just be the original function?

What is it's solution?

Solving a DE means that you correctly figure out what expression makes the equation true.Is solving a differential equation just another way of integrating?

This could amount to integrating - but need not involve the formal process of solving an integration.

You'll see what I mean as you move on to more complicated DEs.

- #7

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As far as solving differential equations goes, there isn't always integration. A lot of them are solved by just guessing the form of the solution and then getting the constants so that it satisfies the initial conditions.

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