# Thread: Implicit Form

1. ## Implicit Form

Completly stuck and do not have a clue, please help! I need to find in implicit form the general solution of the differential equation:

$\displaystyle \frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}$

This is then followed by finding the corresponding particular solution (in implicit form) that satisfies the initial condition:

y=1/6 when x=0

Many thanks

2. $\displaystyle \frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}$

$\displaystyle \frac{dy}{y^2} = 3e^{-2x}\sqrt{8+e^{-2x}} \, dx$

$\displaystyle -\frac{1}{y} = -\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C$

can you finish up?

I've come up with this solution to finish it, could you please check it and if it's wrong please show me where!

From where you left me:

8y dy/dx = (e^-2x)^3/2dx

y8=2/x-c

y=4/3 x=0

Many thanks

4. $\displaystyle \frac{dy}{dx} = 3y^2e^{-2x}\sqrt{8+e^{-2x}}$

$\displaystyle \frac{dy}{y^2} = 3e^{-2x}\sqrt{8+e^{-2x}} \, dx$

$\displaystyle -\frac{1}{y} = -\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C$

$\displaystyle y = \frac{1}{\left(8 + e^{-2x}\right)^{\frac{3}{2}} + C}$

$\displaystyle y(0) = \frac{1}{6}$

$\displaystyle \frac{1}{6} = \frac{1}{27 + C}$

$\displaystyle C = -21$

$\displaystyle y = \frac{1}{\left(8 + e^{-2x}\right)^{\frac{3}{2}} - 21}$