Separable equations: 1st order DE

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In summary, The first step in solving the equation dy/dx = (4x - x^3) / (4 + y^3) is to rewrite it as (4+y^3)dy = (4x-x^3)dx. This is done by separating the variables and moving the y terms from the denominator to the numerator. This is necessary because of the property that if A/B = C/D, then AD = BC.
  • #1
jenzao
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Homework Statement


solve the quation dy/dx = (4x - x^3) / (4 + y^3)
the first thing the book does is rewrite the equation as:

(4+y^3)dy = (4x-x^3)dx

and i understand that they are 1st separating it out... BUT shouldn't it be (1 / (4+y^3))dy?

How can they dissmiss the fact that the y terms are in the denominator?
On every problem, the fact that terms are under denominator gets ignored --why?

Homework Equations


thanks!


The Attempt at a Solution


 
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  • #2
[tex]\frac{dy}{dx} = \frac{4x - x^3}{4 + y^3}[/tex]

[tex]\times (4 + y^3)[/tex]


[tex](4 + y^3) \frac{dy}{dx} = 4x - x^3[/tex]


Now separate the variables.
 
  • #3
jenzao said:
How can they dissmiss the fact that the y terms are in the denominator?

Hi jenzao! :smile:

The y terms are in the denominator on the RHS,

but when you move them over to the LHS, they must go on top.

(and the dx on the bottom of the LHS must go on the top of the RHS for the same reason)

Technically, that's because if A/B = C/D, then AD = BC. :smile:
 

1. What is a separable equation?

A separable equation is a type of first-order differential equation that can be written in the form dy/dx = f(x)g(y), where f(x) and g(y) are functions that only depend on one variable. This type of equation can be solved by separating the variables and integrating both sides.

2. How do you solve a separable equation?

To solve a separable equation, you first separate the variables by moving all the terms containing y to one side of the equation and all the terms containing x to the other side. Then, you can integrate both sides with respect to their respective variables and include a constant of integration. Finally, you can solve for y to get the general solution.

3. What is the difference between a separable equation and a non-separable equation?

A separable equation can be written in the form dy/dx = f(x)g(y), while a non-separable equation cannot be written in this form. This means that a separable equation can be solved using the method of separation of variables, while a non-separable equation requires other methods to solve.

4. Can all first-order differential equations be solved using separation of variables?

No, not all first-order differential equations can be solved using separation of variables. Only equations that can be written in the form dy/dx = f(x)g(y) are separable. Other methods, such as the method of integrating factors, may be needed to solve other types of first-order differential equations.

5. Are there any applications of separable equations in real life?

Yes, separable equations have many applications in various fields of science and engineering. For example, they can be used to model population growth, chemical reactions, and electrical circuits. They are also commonly used in physics to describe the motion of objects subject to a constant force.

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