Separable Differential Equations

In summary, the steps for solving a differential equation in the form \frac{dy}{dx} = h(x)g(y) are valid because dividing by g(y) allows for integration and the result can be obtained by using the chain rule. These steps may seem incorrect, but they are commonly used and considered intuitive.
  • #1
I have read that, if you given a differential equation [itex]\frac{dy}{dx} = f(x,y)[/itex], and can write it in the form [itex]\frac{dy}{dx} = h(x)g(y)[/itex], then you can proceed with the following steps:

[itex]\frac{dy}{g(y)} = h(x)dx[/itex]


[itex]G(y) = H(x) + c[/itex]

Why are these steps vaild? I thought that one was not supposed to regard [itex]\frac{dy}{dx}[/itex]. I have heard that you can regard it as a fraction, because, before taking the limit, you can manipulate the fraction [itex]\frac{\Delta y}{\Delta x}[/itex].

Could someone please help?
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  • #2
The justification is that dividing by g(y) results in

[tex]\frac{\mathrm{d}y}{\mathrm{d}x}=h(x)g(y)\implies h(x)=\frac{1}{g(y)}\frac{ \mathrm{d} y}{\mathrm{d}x}[/tex]

which then, supposing (chain rule) that for some function G(y)

can be integrated to

[tex]\int h(x) \mathrm{d} x = \int \frac{1}{g(y)}\frac{\mathrm{d}y}{\mathrm{d}x}\mathrm{d}x = \int (\frac{\mathrm{d}}{\mathrm{d}x}G(y)) \mathrm{d}x= G(y) = \int (\frac{\mathrm{d}}{\mathrm{d}y}G(y))\mathrm{d}y=\int \frac{1}{g(y)} \mathrm{d}y,[/tex]

the result you get if you "divide by dy". The steps you made are commonly used as they are often considered more intuitive.
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  • #3
You can think of the steps that seem incorrect as mnemonics to obtain a correct result. The justification is the one given above.

1. What is a separable differential equation?

A separable differential equation is a type of differential equation in which the variables can be separated and solved independently. This means that the equation can be written in a form where the variables are on separate sides of the equation, making it easier to solve.

2. How do you solve a separable differential equation?

To solve a separable differential equation, you must first separate the variables and then integrate both sides of the equation separately. This will result in an equation with only one variable on each side. From there, you can solve for the variables and find the solution to the differential equation.

3. What are the applications of separable differential equations?

Separable differential equations are used to model a wide range of physical and biological phenomena, such as population growth, chemical reactions, and radioactive decay. They are also used in engineering and physics to describe systems that change over time.

4. Can separable differential equations have more than one solution?

Yes, separable differential equations can have multiple solutions. When solving a separable differential equation, it is important to check that each solution satisfies the original equation. If there are multiple solutions that satisfy the equation, they are all considered valid solutions.

5. Are there any limitations to using separable differential equations?

While separable differential equations can be applied to a wide range of problems, they are limited in their ability to accurately model complex systems. Additionally, some differential equations may not be separable and require different methods of solving. It is important to consider the limitations of separable differential equations when applying them to a problem.

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