First order differential equation

[Romaha_1]

Homework Statement

Solve the differential equation:

Homework Equations

1+(x-x^2*e^(2y))(dy/dx) = 0

The Attempt at a Solution

No idea how to approach this.

 

[HallsofIvy]

Homework Statement

Solve the differential equation:

Homework Equations

1+(x-x^2*e^(2y))(dy/dx) = 0

The Attempt at a Solution

No idea how to approach this.

It looks pretty clear to me. Basic algebra gives
x(1- e^{2y})dy/dx= -1
and then separate variables as
(1- e^{2y})dy= -\frac{1}{x}dx

Now integrate.

 

[JFonseka]

It looks pretty clear to me. Basic algebra gives
x(1- e^{2y})dy/dx= -1
and then separate variables as
(1- e^{2y})dy= -\frac{1}{x}dx

Now integrate.

Hmm, there's an x^2 in the equation, I think you missed that bit.

 

[tiny-tim]

Hi Romaha_1! Welcome to PF!

(try using the X² tag just above the Reply box )

Does putting z = e^2y help?

 

[Altabeh]

Hi Romaha_1! Welcome to PF!

(try using the X² tag just above the Reply box )

Does putting z = e^2y help?

No it doesn't, does it?

 

[Romaha_1]

Hmm, there's an x^2 in the equation, I think you missed that bit.

Yes, that's the reason I don't know how to do it. I don't think it is a mistake in the problem, since there is a sign "(very tricky)" after it on the problem set.

 

[tiny-tim]

No it doesn't, does it?

Doesn't it?

 

[JFonseka]

Yes, that's the reason I don't know how to do it. I don't think it is a mistake in the problem, since there is a sign "(very tricky)" after it on the problem set.

The question isn't necessarily wrong, I'm not too good at this stuff at all. It just seemed to me that HallsofIvy was attempting the separable ODE approach, but missed the x^2.

I think it should be:

(e^2y)*dy = ((-1-x)/(-x^2))*dx

(e^{2y})*dy = (\frac{1-x}{-x^{2}})*dx

after the separation.

Now you must integrate it. I'm not too sure if I'm right, so be careful.

 

[Romaha_1]

The question isn't necessarily wrong, I'm not too good at this stuff at all. It just seemed to me that HallsofIvy was attempting the separable ODE approach, but missed the x^2.

Yeah, I understand what you meant and agree with you that the separable approach does not work; it was my assumption that the question was wrong.

 

[JFonseka]

Yeah, I understand what you meant and agree with you that the separable approach does not work; it was my assumption that the question was wrong.

I never said the Separable ODE approach doesn't work.

 

[Romaha_1]

I never said the Separable ODE approach doesn't work.

Sorry, for some reason I could not see the end of your previous post. But still, I do not understnad how this works because the whole expression (x-x²*e^2y) is multiplied by dy/dx, not just x²*e^2y.

 

[Romaha_1]

Hi Romaha_1! Welcome to PF!

(try using the X² tag just above the Reply box )

Does putting z = e^2y help?

Hi tiny-tim, thank you for the idea!

But I could solve it only when letting z = x²*e^2y; I still do not see how this is possible with z = e^2y.

 

[tiny-tim]

No idea how to approach this.

Does putting z = e^2y help?

Hi tiny-tim, thank you for the idea!

But I could solve it only when letting z = x²*e^2y; I still do not see how this is possible with z = e^2y.

ah … on this forum, we don't give you the full answer …

so, since you had no idea, I sort-of pushed you half-way (-1 = (x - x²z)(1/2z)dz/dx), and left you to finish it.

having said that, I don't see how z = x²*e^2y does it … perhaps I'm misreading the question?