Do you have a question?Dusty912 said:Homework Statement
Solve each of the following differential equations:
4xydx + (x2 +1)dy=0Homework Equations
None
The Attempt at a Solution
4xydx + (x2 +1)dy=0
(x2 +1)dy=-4xydx
dy/y=-(4xdx)/(x2 +1)
∫dy/y=∫-(4xdx)/(x2 +1)
ln|y|=-2ln|x2+1| +C
used u-sub on last step fo u=x2 +1
A separable differential equation is a type of differential equation where the dependent variable and the independent variable can be separated on opposite sides of the equation. This allows the equation to be solved by integrating both sides separately.
To solve a separable differential equation, you first need to rearrange the equation so that the dependent variable and independent variable are on opposite sides. Then, you can integrate both sides separately and solve for the constant of integration. Finally, you can substitute the constant back into the equation to find the solution.
The main difference between a separable differential equation and a non-separable differential equation is that in a separable equation, the dependent and independent variables can be separated on opposite sides of the equation, whereas in a non-separable equation, they cannot be separated in this way. This makes a separable equation easier to solve because it can be solved by integrating both sides separately.
Separable differential equations have many real-world applications in various fields of science and engineering. Some examples include population growth models, chemical reactions, and predator-prey relationships in ecology.
There are a few strategies that can be helpful when solving difficult separable differential equations. One strategy is to try to simplify the equation by factoring, using trigonometric identities, or making a substitution. Another strategy is to use separation of variables, where you separate the dependent and independent variables into different equations and then solve for each separately. It is also important to carefully check your work and make sure you have correctly integrated and solved for the constant of integration.