Finding a General Solution

**Deadstar** · January 21st 2008, 01:28 PM

Not sure if this is the right place to post this cos its off my module called Applied Maths but seemed more of a calculus topic...

$y^{''} + y^{'} + 2y^2 (1 - y) = 0$

Find the value of the constant a so that $y = (1 + e^{ax})^2$ is a solution of the above equation.

Now i just plugged the y = ... bit into the equation but it ended as

... $= \frac{1}{(1 + e^{ax})^2} (1 - a^2 e^{ax} - \frac{2}{(1 + e^{ax})} + \frac{2a^2 e^{2ax}}{(1 + e^{ax})})$
and figured something had gone wrong somewhere as i cant solve it from there.

Any help appreciated

**Peritus** · January 21st 2008, 01:56 PM

it's a lot to write...

$ \begin{gathered} y'' + y' + 2y^2 (1 - y) = 0 \hfill \\ y_{solution} = \left( {1 + e^{ax} } \right)^2 \hfill \\ \end{gathered} $

$ \begin{gathered} y' = 2ae^{ax} \left( {1 + e^{ax} } \right) \hfill \\ y'' = 2a^2 e^{ax} \left( {1 + 2e^{ax} } \right) \hfill \\ \end{gathered} $

substituting these derivatives into the equation:

$ 2a^2 e^{ax} \left( {1 + 2e^{ax} } \right) + 2ae^{ax} \left( {1 + e^{ax} } \right) + 2\left( {1 + e^{ax} } \right)^4 \left[ {1 - \left( {1 + e^{ax} } \right)^2 } \right] = 0 $

$ 2a^2 e^{ax} \left( {1 + 2e^{ax} } \right) + 2ae^{ax} \left( {1 + e^{ax} } \right) + 2\left( {1 + e^{ax} } \right)^4 \left[ {1 - \left( {1 + e^{ax} } \right)} \right]\left[ {1 + \left( {1 + e^{ax} } \right)} \right] = 0 $

$ 2a^2 e^{ax} \left( {1 + 2e^{ax} } \right) + 2ae^{ax} \left( {1 + e^{ax} } \right) + 2\left( {1 + e^{ax} } \right)^4 \left[ { - e^{ax} } \right]\left[ {2 + e^{ax} } \right] = 0 $

$ 2e^{ax} \left[ {a^2 + 2a^2 e^{ax} + a + ae^{ax} - \left( {1 + e^{ax} } \right)^4 \left( {2 + e^{ax} } \right)} \right] = 0 $

now the outside exponent can nulify the equation only if a->inf which is not interesting.

$ a^2 + a + \left( {2a^2 + a} \right)e^{ax} = \left( {1 + e^{ax} } \right)^4 \left( {2 + e^{ax} } \right) $

now because the following equality is supposed to hold for any value of x (y is a solution) then it must hold in particular for x = 0 thus:

$ \begin{gathered} a^2 + a + 2a^2 + a = 48 \hfill \\ \hfill \\ 3a^2 + 2a - 48 = 0 \hfill \\ \hfill \\ ... \hfill \\ \end{gathered} $