differential eq solution

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Mar 4th 2008, 07:26 AM
dankelly07

differential eq solution

Just wondering if anyone could have a look through nd see if I've done this right? When it comes to the general solution & conditon I'm not sure if I've done this right.

$ \begin{gathered} y' - 2xy = x \hfill \\ y' = 2xy + x \hfill \\ \smallint \frac{{dy}} {y} = \smallint 3x \hfill \\ \log (y) = \frac{3} {2}x^2 \hfill \\ y(x) = A\exp \{ \frac{3} {2}x^2 \} \hfill \\ \end{gathered} $

satisfies

$ y(0) = 0 $

is the answer y(x)=A??
I think I might have gone wrong somewhere
Mar 4th 2008, 07:32 AM
kalagota

Quote:

Originally Posted by dankelly07

Just wondering if anyone could have a look through nd see if I've done this right? When it comes to the general solution & conditon I'm not sure if I've done this right.

$ \begin{gathered} y' - 2xy = x \hfill \\ y' = 2xy + x \hfill \\ \smallint \frac{{dy}} {y} = \smallint 3x \hfill \\ \log (y) = \frac{3} {2}x^2 \hfill \\ y(x) = A\exp \{ \frac{3} {2}x^2 \} \hfill \\ \end{gathered} $

satisfies

$ y(0) = 0 $

is the answer y(x)=A??
I think I might have gone wrong somewhere

yes.. on the third line
check your factoring..
it should be $y' = (2y + 1)x$ right?
Mar 4th 2008, 08:40 AM
dankelly07

so..

$ \smallint \frac{{dy}} {{2y + 1}} = \smallint xdx $

$ \frac{1} {2}\log (|2y + 1|) = \frac{{x^2 }} {\begin{gathered} 2 \hfill \\ \hfill \\ \end{gathered} } + A $

$ 2y = e^{-x^2 } - 1 + A $

$ y = A + \frac{{e^{-x^2 } - 1}} {\begin{gathered} 2 \hfill \\ \hfill \\ \end{gathered} } $

am I close?
Mar 5th 2008, 05:47 AM
kalagota

$\frac{1}{2} \log (2y + 1) = \frac{x^2}{2} + A \Rightarrow \log (2y+1) = x^2 + A$

if you take the exponential of both sides, it should turn out to be like this..

$2y+1 = e^{x^2 + A} = Ae^{x^2}$ and not the one have written.. continue now..
Mar 6th 2008, 03:29 PM
dankelly07

thanks for your help..

so

Ae^(x^2)+1/2

with condition A=1??

I want to be sure I can do this..
Mar 6th 2008, 08:34 PM
mr fantastic

Quote:

Originally Posted by dankelly07

thanks for your help..

so

Ae^(x^2)+1/2 Mr F says: It's - not + 1/2.

with condition A=1??

I want to be sure I can do this..

$2y + 1 = A e^{x^2}$

$\Rightarrow y = Ae^{x^2} {\color{red} - }\frac{1}{2}$ .

Substitute the given boundary condition y(0) = 0, that is, y = 0 when x = 0:

$0 = A - \frac{1}{2} \Rightarrow A = \frac{1}{2}$ .

Therefore $y = \frac{1}{2} e^{x^2} - \frac{1}{2}$ .

You should confirm this solution by substitution into the DE.