# How To Find The General Solution?

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• April 11th 2010, 02:34 PM
AlphaRock
How To Find The General Solution?
1. How do you find the general solution for the given differential equation?
dP/dt = P(P+3)

Answer: P = (3Ae^(3t))/(1-Ae^(3t))

2. Of the pine trees in Manning Park, 400 000 are infested with a fungus. If aerial spraying with 400 kg/km^2 of a fundicide will reduce the number of trees infested to 40 000, how much is needed per square kilometre to reduce the number of trees infested to 2 000? (assume an exponential relationship.)

Answer: 920.4 kg/km^2
• April 11th 2010, 02:47 PM
alexmahone
$\frac{dP}{dt}=P(P+3)$

$\frac{dP}{P(P+3)}=dt$

$\frac{dp}{P}-\frac{dp}{P+3}=3dt$

Integrating both sides,

$ln P-ln (P+3)=3t+C$

$ln \frac{P}{P+3}=3t+C$

$\frac{P}{P+3}=e^{3t}e^C$

$\frac{P}{P+3}=Ae^{3t}$

$P=PAe^{3t}+3Ae^{3t}$

$P(1-Ae^{3t})=3Ae^{3t}$

$P=\frac{3Ae^{3t}}{1-Ae^{3t}}$