# Thread: Stuck trying to get a general solution for a separable differential equation

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

Hi, I have the following problem:

Consider the differential equation

$\displaystyle dy/dx=y^2+3y+2$

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)

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

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

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

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

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

$\displaystyle ln|y+1|-ln|y+2|=x+C$

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

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

So that's where I am stuck

2. $\displaystyle \displaystyle \frac{y+1}{y+2} = \frac{y+2-1}{y+2}= \frac{y+2}{y+2}- \frac{1}{y+2}= 1- \frac{1}{y+2}$