A question about seperable differential equations

In summary, the equation y-1\frac{dy}{dx} = x can be manipulated to show that dy and dx are operators that can be manouvred, and that they are not relevant to solving differential equations.
  • #1
Gregg
459
0
[itex]\frac{dy}{dx} = \frac{x}{y-1}[/itex]

You separate it

[itex] (y-1)\frac{dy}{dx} = x [/itex]

Some people have moved the 'dx' to the other side and manipulated the dy/dx as if it's just a part of the equation that can be just moved around and that 'dy' and 'dx' are some sort of operators that can be manouvred like this. Others, when they integrate both sides with respect to x show that the 'dx's cancel out on one side leaving with respect to y. And someone said that it can be changed to an integral on the LHS due to the chain rule.

All of these are understandable but I cannot see them to be very logical and I don't know of any way I can manipulate 'dy's and 'dx's and for them to be of any use so could someone explain why,

[itex] (y-1)\frac{dy}{dx} = x [/itex],

[itex]\therefore \int (y-1)\frac{dy}{dx} dx = \int x dx [/itex],

[itex]\therefore \int (y-1) dy =\int x dx [/itex],

whether singular dy and dx hold any relevance in solving differential equations and further whether I should attempt to isolate them if so.
 
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  • #2
I don't know much reasoning but i know how to do this.
What i would do is move dy with the y terms and dx with x term. then you could intergre both side and get 1/2(y^2)-y=1/2 x^2
 
  • #3
Separating the dy and dx is not relevant to solving the problem and is just notational. The theorem being used is that if a(x) = b(x), then the antiderivative of a should be equivalent to the antiderivative of b up to a constant.
In particular,
[tex]\int (y-1)\frac{dy}{dx} dx[/tex]
is evaluated using the change of variable theorem:
[tex]\int_a^b f(y(x))*y'(x) dx[/tex]
[tex]= \int_{y(a)}^{y(b)} f(y) dy[/tex]
 
Last edited:

Related to A question about seperable differential equations

1. What is a separable differential equation?

A separable differential equation is a type of differential equation where the variables can be separated into two functions, one containing the dependent variable and the other containing the independent variable. This allows for the equation to be solved by integrating both sides separately.

2. How do I know if a differential equation is separable?

A differential equation is separable if it can be written in the form of M(x)dx = N(y)dy, where M(x) and N(y) are functions of x and y respectively. If the equation can be rearranged to this form, then it is separable.

3. What is the process for solving a separable differential equation?

The process for solving a separable differential equation involves separating the variables, integrating both sides, and then solving for the constant of integration. This will result in the general solution, which can then be used to find the particular solution by plugging in the initial conditions.

4. Can I solve a separable differential equation without integration?

No, integration is necessary in order to solve a separable differential equation. This is because the variables must be separated in order to solve the equation, and integration is the mathematical process used to combine the two separate functions.

5. What are some applications of separable differential equations in science?

Separable differential equations are commonly used in various fields of science, such as physics, chemistry, and engineering. They can be used to model and solve problems involving rates of change, growth and decay, and other dynamic systems. Examples include population growth, radioactive decay, and chemical reactions.

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