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Could you, for instance, replace [itex]\frac{dy}{dx}[/itex] with [itex]y'(x)[/itex] in a differential equation and work it out?

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- Thread starter jaydnul
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In summary: It's always helpful to get a second opinion!In summary, Liebnitz's notation is just a compact way to write differential equations, and you can solve them using either primed or prime notation.

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Could you, for instance, replace [itex]\frac{dy}{dx}[/itex] with [itex]y'(x)[/itex] in a differential equation and work it out?

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- #2

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You can treat the Liebnitz notation as a fraction because that's the way Liebnitz worked out the notation. In that sense it is not "just" a notation.

You can use the primed notation - many do - it's just harder to make the relations.

You can solve, for eg. ##y'(x)=a## by integrating both sides wrt x: $$\int y'\cdot dx = a\int dx \Rightarrow y=ax+c$$

But what about ##y'(x)=ay## ?

You can use the primed notation - many do - it's just harder to make the relations.

You can solve, for eg. ##y'(x)=a## by integrating both sides wrt x: $$\int y'\cdot dx = a\int dx \Rightarrow y=ax+c$$

But what about ##y'(x)=ay## ?

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- #3

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[tex]∫ydx=e^x+c[/tex]

So how would I solve this using prime notation:

[tex]y'(x)=y[/tex]

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I had hoped you'd do something like this: $$\frac{dy}{dx}=ay\\ \Rightarrow \int \frac{dy}{y}=a \int dx \\ \Rightarrow \ln |y|=ax+c \\ \Rightarrow y=e^{ax+c}=e^ce^{ax}=Ce^{ax}$$Jd0g33 said:Using [itex]\frac{dy}{dx}[/itex] notation, I get:

[tex]∫ydx=e^x+c[/tex]...

That's my point: there's no systematic way like there is with Leibnitz - you'd have to recognize the form of the equation and apply a rule or use guesswork.So how would I solve this using prime notation:

[tex]y'(x)=y[/tex]

You'd notice it is of form ##\sum a_n y^{(n)}=0## and

- #5

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So how did Newton do his differential equations?

Btw Simon, I really appreciate the help!

Btw Simon, I really appreciate the help!

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Presumably if he had to use infinitesimal differentials he'd treat them as an inverse integration... using his experience of integration to guide his guesses. Substitute in as above.

Notice: ##\int y'(x)dx = y## right? So integrating both sides of the example he'd have got ##y=\int y(x)dx## and he'd look for equations which have that property. What integrates to give itself? It's just the same as when you do ##\small 3x=12## - you are asking what, multiplied by 3, gives you 12. You do it by experience and look-up tables (which you have memorized).

Note: The number one main method for solving DEs remains the guesswork method.

We just know a lot of tricks for figuring out a good guess.

- #7

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Ya that makes sense because the only function of x that satisfies [itex]y=y'[/itex] and [itex]y=∫y[/itex] would be [itex]e^x[/itex].

I can see how that would get complicated haha.

Thanks again Simon!

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Newtons notation is compact though, and easier to type, so you'll still see it a lot for written notes.

No worries - see that "thanks" button?

A differential equation is an equation that relates one or more unknown functions to their derivatives. It is used to model various physical, biological, and economic phenomena in fields such as physics, engineering, and economics.

An ordinary differential equation involves only one independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations are commonly used to model systems that change over time, while partial differential equations are used to model systems that vary in space and time.

The most commonly used notations for differential equations are the Newtonian notation, the Leibniz notation, and the Lagrange notation. The Newtonian notation uses a dot above the variable to represent the derivative, while the Leibniz notation uses d/dx to represent the derivative. The Lagrange notation uses prime symbols (') to represent the derivative.

The method for solving a differential equation using different notations depends on the type of differential equation and the specific notation used. In general, the first step is to identify the type of differential equation (ordinary or partial) and the order (highest derivative present). Then, the appropriate method, such as separation of variables or substitution, can be used to solve the equation using the chosen notation.

Differential equations have numerous real-life applications, including modeling population growth, predicting stock market fluctuations, and designing control systems for machines and vehicles. They are also used in fields such as fluid dynamics, chemistry, and biology to understand and predict natural phenomena.

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