calculus - population model/de

**mathshelpbook** · May 11th 2008, 01:29 AM

having so much trouble with de's in general! i think it's a linear de, but really stuck! please help:

dp/dt = -50 + 15p - p^2

thanks in advance!

**flyingsquirrel** · May 11th 2008, 02:03 AM

Hello

There is a $p^2$ in the expression so the equation is not linear. However, the two variables $p$ and $t$ can be separated : $\frac{\mathrm{d}p}{p^2-15p+50}=-\mathrm{d}t$ and we'll get the solution by integrating the both sides.

To achieve this, one needs to make appear a derivative on the LHS. As the denominator contains a $p^2$ , one can try to transform $\frac{\mathrm{d}p}{p^2-15p+50}$ into $\mathrm{constant}\cdot\frac{\mathrm{d}u}{1+u^2}$ which anti-derivative is $\mathrm{constant}\cdot \arctan u$ . Hence solving this problem boils down to finding $u$ . Do you have any idea on how this can be done ?

**mathshelpbook** · May 11th 2008, 02:26 AM

thanks flyingsquirrel.
well i have no idea, but i thought maybe i could try 'flipping' it as the rhs is in terms of p

ie. dt/dp = - 1/(p^2 - 15p + 50)
and then t = - integral[1/(p^2 - 15p + 50)] dp

does that work? i'm not sure if that is how to start the question

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