Could someone show me the steps to separating the variables in the following differential equation:
dy/dx = (2x+1) / (y+1)? I keep messing it up.
To separate:
dy/dx = (2x+1) / (y+1)
Cross multiply:
(y+1)dy = (2x+1)dx
To solve, integrate both sides
(y^2/2+y) = (x^2+x)+C
where C is an integration constant
Thank you! How would I isolate y by itself to make this a y= expression?
This would be more messy if you are looking for an explicit solution.
You would solve for y as a quadratic equation, and verify that all solutions obtained from the equation are actual solutions.
Most of the time, an implicit solution (where y occurs in more than one term or embedded as a function) is acceptable.
Certainly! To separate variables in a differential equation, you need to arrange it such that all the terms involving y are on one side of the equation, while all the terms involving x are on the other side.
In the given equation:
dy/dx = (2x + 1) / (y + 1)
To separate the variables, we first multiply both sides of the equation by (y + 1):
(y + 1) dy = (2x + 1) dx
Now you have separated the variables: dy on one side and dx on the other side.
Next, you can integrate both sides of the equation. On the left side, you integrate with respect to y, and on the right side, you integrate with respect to x.
∫ (y + 1) dy = ∫ (2x + 1) dx
Integrating the left side with respect to y, you get:
(1/2) y^2 + y = ∫ (2x + 1) dx
Integrating the right side with respect to x, you get:
(1/2) y^2 + y = x^2 + x + C
Here, C represents the constant of integration that arises during the integration process.
So, after separating and integrating the variables, the equation becomes:
(1/2) y^2 + y = x^2 + x + C
This is the final result after separating the variables and integrating both sides of the equation.