Solving differential equations with different differential notation?

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.
  • #1
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Since [itex]\frac{dy}{dx}[/itex] is just considered notation, how can we treat it as an actual fraction when soliving differential equations?

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
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## ?
 
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  • #3
Using [itex]\frac{dy}{dx}[/itex] notation, I get:
[tex]∫ydx=e^x+c[/tex]

So how would I solve this using prime notation:
[tex]y'(x)=y[/tex]
 
  • #4
Jd0g33 said:
Using [itex]\frac{dy}{dx}[/itex] notation, I get:
[tex]∫ydx=e^x+c[/tex]...
I had hoped you'd do something like this: $$\frac{dy}{dx}=ay\\ \Rightarrow \int \frac{dy}{y}=a \int dx \\ \Rightarrow \ln |y|=ax+c \\ \Rightarrow y=e^{ax+c}=e^ce^{ax}=Ce^{ax}$$
So how would I solve this using prime notation:
[tex]y'(x)=y[/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.

You'd notice it is of form ##\sum a_n y^{(n)}=0## and guess ##y=e^{\lambda x}## and substitute the guess into the equation to find lambda.
 
  • #5
So how did Newton do his differential equations?

Btw Simon, I really appreciate the help!
 
  • #6
Newton's contribution was primarily in integration, which he could verify geometrically. If you look through Principia, for eg. you'll see he didn't use calculus for any of his proofs.

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
Oh I see.

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!
 
  • #8
Oh sure - DEs get arbitrarily complicated even at 1st order.
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?
 
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1. What is a differential equation?

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.

2. What is the difference between ordinary and partial differential equations?

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.

3. What are the different notations used in solving differential equations?

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.

4. How do I solve a differential equation using the different notations?

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.

5. What are some real-life applications of solving differential equations?

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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