Can You Solve This Non-Exact Differential Equation?

[freezer]

Homework Statement

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

Homework Equations

The Attempt at a Solution

After messing with this several times, i went through the through the textbook 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.

<br /> <br /> \frac{\partial F}{\partial y}= y + 2e^x \neq \frac{\partial F}{\partial x} = e^x<br />

 

[rock.freak667]

Since I am seeing that y+2e^x - e^x = y+e^x which is the same as one of your terms, you might need to use an integrating factor for a DE that is not exact.

You can see here for the integrating factor formula

http://www.sosmath.com/diffeq/first/intfactor/intfactor.html

 

[freezer]

I see what you are saying but not how you can simplify the origional eq into what you got.

I get

<br /> y(\frac{y}{2}+2e^x) + (y+e^x)y&#039; = 0<br />

if you say that

p(x) = (\frac{y}{2}+2e^x)

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

If i distribut the y'

<br /> <br /> \frac{y^2}{2}+2ye^x + yy&#039;+ y&#039;e^x = 0<br /> <br />

rearange a bit

<br /> \frac{y^2}{2}+ yy&#039; + 2ye^x + + y&#039;e^x = 0<br />

factor

<br /> y(\frac{y}{2}+ y&#039;) + e^x(2y + y&#039;) = 0<br />

but then what?

 

[freezer]

I was thinking:

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

Then:

<br /> <br /> \frac{xy^2}{2}+ 2ye^x = -\frac{1}{2}y^2 - ye^x + c<br />

then to:
<br /> \frac{xy^2}{2}+ 2ye^x + \frac{1}{2} y^2 + ye^x = c<br />

 

[jackmell]

I was thinking:

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

Then:

<br /> <br /> \frac{xy^2}{2}+ 2ye^x = -\frac{1}{2}y^2 - ye^x + c<br />

You can't integrate like that since y is a function of x. You need to put it into standard form:

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

Then:

\frac{1}{N}(M_y-N_x)=1=f(x)

The integrating factor is then e^{\int 1 dx}

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.

 

[HallsofIvy]

I did not know that Dixie Queen did this kind of calculation!