First Order Separable Differential Equation

[Roadrunner2015]

(1+y²)(e^^2xdx-e^ydy)-(1+y)dy = 0

I know this is separable but I honestly don't know where to begin to solve this. If someone could help me through the steps or give me some hints on where to start it would be very much appreciated. Thank you very much.

 

[Deleted member 4993]

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

I know this is separable but I honestly don't know where to begin to solve this. If someone could help me through the steps or give me some hints on where to start it would be very much appreciated. Thank you very much.

Start by separating the variables - i.e. - terms attached to dy on side of "=" and terms attached to dx on the other side of the "=".

 

[Roadrunner2015]

e^2xdx = (1+y)/(1+y²)dy + e^ydy

Would that be correct in separating the terms?

Also, is it appropriate to write that as

e^2xdx = ( ((1 + y)/(1 + y²)) + e^y )dy

or no, because I will need to integrate anyways and that would just separate them again?

Also, would the next step be to integrate or am I missing something?


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Possible answer being

1/2e^2x = 1/2ln(y² + 1) + tan^-1y + e^y

1/2ln(y² + 1) + tan^-1y + e^y - 1/2e^2x = C

?

 

[Deleted member 4993]

(1+y²)(e^^2xdx-e^ydy)-(1+y)dy = 0

$\displaystyle \displaystyle{(1 + y^2) * \left [e^{2x} dx - e^y dy \right ] \ \ = \ (1 + y) dy}$

$\displaystyle \displaystyle{e^{2x} \ \ dx \ = \ \left [\frac{1 + y }{1 + y^2} + e^y \right ] dy}$

 

[Roadrunner2015]

Could you possibly explain how you got e^y into the numerator?

When I was attempting to sort the variables, moving e^y was the last step as if I tried to move it before that, I couldn't because it was being multiplied by (1+y²).

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The steps I took were:

(1+y²)(e^2xdx - e^ydy) - (1+y)dy = 0

1) Moving - (1+y)dy to the right hand side. Making the equation (1+y²)(e^2xdx - e^ydy) = (1+y)dy

2) Dividing (1+y²) from both sides. Making the equations e^2xdx - e^ydy = (1+y)/(1+y²)dy

3) Then adding e^ydy to both sides to make the equation e^2xdx = (1+y)/(1+y²)dy + e^ydy

 

[Deleted member 4993]

Could you possibly explain how you got e^y into the numerator?

When I was attempting to sort the variables, moving e^y was the last step as if I tried to move it before that, I couldn't because it was being multiplied by (1+y²).

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The steps I took were:

(1+y²)(e^2xdx - e^ydy) - (1+y)dy = 0

1) Moving - (1+y)dy to the right hand side. Making the equation (1+y²)(e^2xdx - e^ydy) = (1+y)dy

2) Dividing (1+y²) from both sides. Making the equations e^2xdx - e^ydy = (1+y)/(1+y²)dy

3) Then adding e^ydy to both sides to make the equation e^2xdx = (1+y)/(1+y²)dy + e^ydy

I made a mistake in reading your post. I corrected it above.

Your steps are correct

 

[Roadrunner2015]

Thank you very much, your posts have helped me finish the problem. Again, thank you very much for all your help.