First order differential equation

In summary: The Attempt at a SolutionGiven that the equation is already solved but i didn't understand one of the steps, if the solutions i'll show you where i get stuck.So you substitute y=tx and here comes the boom, by doing this substitution they say the result is -\frac{dx}{x} and from here on it's quite simple, you just integrate both sides and get the result.But i didn't understand the steps from substituting y=xt to getting to that result.I'll show you how i tried to solve it but got to a different result.So if y=tx --> dy/dx=tdx and it's clear that if you substitute
  • #1
squareroot
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Homework Statement


I'm starting college this autumn(physics) and I started learning some calculus on my own, basic stuff like first order differential equation and so on.Recently i stumbled on something that i don t understand.I was reading the course and re-solving the given examples for myself.So you have the equation

[tex] \frac{dy}{dx} = \frac{ \frac{y}{x} - 1 }{ \frac{y}{x} + 1 } [/tex]

Homework Equations



calculus

The Attempt at a Solution


Given that the equation is already solved but i didn't understand one of the steps if the solutions i'll show you where i get stuck.

So you substitute y=tx and here comes the boom, by doing this substitution they say the result is
[tex] \frac{t+1}{t2+1}dt =-\frac{dx}{x} [/tex] and from here on it s quite simple, you just integrate both sides and get the result.But i didn't understand the steps from substituting y=xt to getting to that result.I ll show you how i tried to solve it but got to a different result.

So if y=tx --> dy/dx=tdx and it s clear that if you substitute dy/dx in that equation you don t get their answer.

My main problem is that, in high school we didn't use this notation, we used f'(x) instead of dy/dx and so on, so I'm having a bit of trouble getting used with this notation, but these are the one used in physics so now i have to squash my brains a little and try to get used to them.

If someone can explain that step to me and maybe give me a few tips on how to make the transition between these two notation easier it would be great!

Thank you
 
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  • #2
"Here comes the boom"?

Seeing the "y/x" on the right, I think it would be obvious to set [itex]t= y/x[/itex] which is the same as [itex]y= xt[/itex]. Differentiating both sides with respect to x,
[tex]\frac{dy}{dx}= x\frac{dt}{dx}+ t[/tex].
And the whole point of that substitution is that
[tex]\frac{\frac{y}{x}- 1}{\frac{y}{x}+ 1}= \frac{t- 1}{t+ 1}[/tex]
So the equation becomes
[tex]x\frac{dt}{dx}+ t= \frac{t- 1}{t+ 1}[/tex]
[tex]x\frac{dt}{dx}= \frac{t-1}{t+ 1}- t= \frac{t- 1}{t+ 1}- \frac{t(t+ 1)}{t+1}[/tex] Combine the fractions on the right and you have a relatively simple "separable" equation.

(Personally, I thing [itex]dy/dx[/itex] is a better notation than y' but you can translate back:
y'= (y/x- 1)/(y/x+ 1). Let t= y/x so that y= xt and y'= xt'+ t. The equation becomes [itex]xt'+ t= (t- 1)/(t+ 1)[/itex]. [itex]xt'= (t- 1)/(t+ 1)- t= (t- 1)/(t+ 1)- (t(t+ 1))/(t+ 1)[/itex]. Of course, when you learned integration, you must have learned that if y'= f(x) then [itex]y= \int f(x)dx[/itex] which follows more easily from [itex]dy/dx= f(x)[/itex], [itex]dy= f(x)dx[/itex].)
 
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  • #3
Ah... Got it! thank you
 

What is a first order differential equation?

A first order differential equation is an equation that contains a function and its derivative. It represents the relationship between a function and its rate of change.

What is the general form of a first order differential equation?

The general form of a first order differential equation is dy/dx = f(x,y), where y is the dependent variable and x is the independent variable.

What is the solution to a first order differential equation?

The solution to a first order differential equation is a function that satisfies the equation. It is usually expressed in terms of the dependent variable, y, in terms of the independent variable, x.

What are the different types of first order differential equations?

There are several types of first order differential equations, including separable, exact, linear, and Bernoulli equations. Each type has its own specific method for solving it.

How are first order differential equations used in science?

First order differential equations are used in many scientific fields, such as physics, chemistry, biology, and engineering. They are used to model and understand various processes, such as population growth, chemical reactions, and motion of objects.

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