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DQ worksheet equation help

  • Thread starter freezer
  • Start date
  • #1
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Homework Statement


This is from a worksheet...
[itex]
(\frac{y^2}{2}+ 2ye^x) + (y + e^x)\frac{_{dy}}{dx} = 0
[/itex]

Homework Equations





The Attempt at a Solution



After messing with this several times, i went through the through the text book for examples but nothing in the chapter matches with the exception of exact.. with a bit of manipulation i matched up the form. But it is not exact so that does not work. We have covered up separable, substitution, and exact and none seem to fit.

[itex]

\frac{\partial F}{\partial y}= y + 2e^x \neq \frac{\partial F}{\partial x} = e^x
[/itex]
 

Answers and Replies

  • #2
rock.freak667
Homework Helper
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  • #3
76
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I see what you are saying but not how you can simplify the origional eq into what you got.

I get

[itex]
y(\frac{y}{2}+2e^x) + (y+e^x)y' = 0
[/itex]

if you say that

p(x) = [itex] (\frac{y}{2}+2e^x) [/itex]

then p(x) is not going to integrate very nice.

If i distribut the y'

[itex]

\frac{y^2}{2}+2ye^x + yy'+ y'e^x = 0

[/itex]

rearange a bit

[itex]
\frac{y^2}{2}+ yy' + 2ye^x + + y'e^x = 0
[/itex]

factor

[itex]
y(\frac{y}{2}+ y') + e^x(2y + y') = 0
[/itex]

but then what?
 
  • #4
76
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I was thinking:

[itex]

(\frac{y^2}{2}+ 2ye^x)dx = -(y+e^x)dy

[/itex]

Then:

[itex]

\frac{xy^2}{2}+ 2ye^x = -\frac{1}{2}y^2 - ye^x + c
[/itex]

then to:
[itex]
\frac{xy^2}{2}+ 2ye^x + \frac{1}{2} y^2 + ye^x = c
[/itex]
 
  • #5
1,796
53
I was thinking:

[itex]

(\frac{y^2}{2}+ 2ye^x)dx = -(y+e^x)dy

[/itex]

Then:

[itex]

\frac{xy^2}{2}+ 2ye^x = -\frac{1}{2}y^2 - ye^x + c
[/itex]
You can't integrate like that since y is a function of x. You need to put it into standard form:

[tex](y^2/2+2ye^x)dx+(y+e^x) dy=0=Mdx+Ndy[/tex]

Then:

[tex]\frac{1}{N}(M_y-N_x)=1=f(x)[/tex]

The integrating factor is then [itex]e^{\int 1 dx}[/itex]

Now multiply both sides of the standard-form by this integrating factor to make it exact then proceed to solve it using the technique of exact equations. Also, first check that it is indeed exact in case I made a mistake.
 
  • #6
HallsofIvy
Science Advisor
Homework Helper
41,808
933
I did not know that Dixie Queen did this kind of calculation!
 

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