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Math Help - Stuck trying to get a general solution for a separable differential equation

  1. #1
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    Stuck trying to get a general solution for a separable differential equation

    Hi, I have the following problem:

    Consider the differential equation

    dy/dx=y^2+3y+2

    A) State whether the differential equation is linear and/or separable.

    B) Obtain the general solution, expressing y explicitly as a function of x.
    ________

    For A) I have that the equation is nonlinear, but it is separable.

    And in B, I am stuck at the end (trying to express y as a function of x).

    I got this, but I don't know what else to do.... thanks! (please, let me know if this is actually a general solution as well)

    dy/dx=1/(y^2+3y+2)^{-1}

    dy(y^2+3y+2)^{-1}=dx

    \int (y^2+3y+2)^{-1}dy=\int dx

    \int 1/(y^2+3y+2)dy=\int dx

    \int 1/(y+1)-\int 1/(y+2) dy=\int dx

    ln|y+1|-ln|y+2|=x+C

    ln|(y+1)/(y+2)|=x+C

    (y+1)/(y+2)=e^{x+C}


    So that's where I am stuck
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  2. #2
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    \displaystyle \frac{y+1}{y+2} = \frac{y+2-1}{y+2}= \frac{y+2}{y+2}- \frac{1}{y+2}= 1- \frac{1}{y+2}
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