CALCULUS

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.

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  1. 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

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  2. Thank you! How would I isolate y by itself to make this a y= expression?

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  3. 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.

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