# Solve using seperation of variables

• Danzilla14
In summary, the conversation discusses solving a differential equation using the method of separation of variables. The individual is having trouble integrating one side of the equation and asks for help. They are advised to factor and use the method of partial fractions.
Danzilla14
Hello everyone how yall durrin!
Solve the following DEs by Seperation of Variables
eliminate natural logarithms and leave your final answer in implicit form

(3x+8)(y^2 +4)dx - 4y(x^2+5x+6)dy=0
by separation of variables i get
(3x+8)dx/(x^2+5x+6)=4ydy/(y^2 +4)
now I am having trouble integrating the left side of the equation, and on the right side i get 2ln|Y^2 +4|
help me pleeeeeease..
preciate it

Factor and then use method of partial fractions.

$$\frac{3x+8}{x^2+5x+6} = \frac{A}{x+2} + \frac{B}{x+3}$$

Take it from there.

Thank you very much kind sir!

You're welcome. Welcome to PF, by the way.

## 1. What is "separation of variables"?

"Separation of variables" is a mathematical technique used to solve differential equations by separating the variables and solving for each one separately.

## 2. How do you use separation of variables?

To use separation of variables, you first need to identify a differential equation that can be solved using this technique. Then, you need to separate the variables by moving all terms with one variable to one side of the equation, and all terms with the other variable to the other side. Finally, you can solve for each variable separately to find the general solution.

## 3. What types of differential equations can be solved using separation of variables?

Separation of variables can be used to solve first-order, homogeneous, and non-homogeneous differential equations. It is most commonly used for solving ordinary differential equations (ODEs).

## 4. Are there any limitations to using separation of variables?

Yes, there are some limitations to using separation of variables. This technique can only be used for certain types of differential equations and may not work for more complex equations. It also assumes that the variables can be easily separated, which may not always be the case.

## 5. Can you provide an example of solving a differential equation using separation of variables?

Yes, for example, to solve the equation dy/dx = 2xy, we can separate the variables by writing it as dy/y = 2x dx. Then, we can integrate both sides to get ln|y| = x^2 + C, where C is a constant. Finally, we can solve for y by taking the exponential of both sides, giving us the general solution of y = Ce^(x^2).

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