# Seperation of Variables Method

#### Benji123

Hi All, new to the forum!

Having a few problems with this question: "Use the separation of variables method to solve the first ordinary differential equation dy/dx = x(y-1) given that when x = -, y = 4

I came up with two answers... 12 and 1.099 and I'm not sure which one is correct, if any of them are correct! Any help will be deeply appreciated, thanks!

dy/dx = x(y-1)

==>

dy/(y-1) = xdx

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

correct?

#### mr fantastic

MHF Hall of Fame
Hi All, new to the forum!

Having a few problems with this question: "Use the separation of variables method to solve the first ordinary differential equation dy/dx = x(y-1) given that when x = -, y = 4

I came up with two answers... 12 and 1.099 and I'm not sure which one is correct, if any of them are correct! Any help will be deeply appreciated, thanks!

#### Benji123

Thanks for the quick replies, I have 2 answers which I got not sure if there correct...

∫(y-1) dy = ∫x dx

y^2 /2 - y = x^2 /2 + C1

Multiplied both sides by 2

y^2 - y + x^2 = C

= 12

Do I need to find 'C' in order to answer this question?

∫1/y-1 dy = ∫x dx

ln(y-1) = x^2 /2 + C1

Mutliplied both sides by 2

ln(y-1) + x^2 = C

= 1.099

Again do I need to find 'C' to answer this question or just leave it with at the integrated solution?

Thanks

Last edited:

#### mr fantastic

MHF Hall of Fame
Thanks for the quick replies, I have 2 answers which I got not sure if there correct...

∫(y-1) dy = ∫x dx Mr F says: This is plainly wrong (at this level I do hope you understand why) and so all that follows from it will be wrong.

y^2 /2 - y = x^2 /2 + C1

Multiplied both sides by 2

y^2 - y + x^2 = C

= 12

Do I need to find 'C' in order to answer this question?

-----------------------------------------------------------------------------------------

∫1/y-1 dy = ∫x dx Mr F says: Post #2 tells you that this is correct.

ln(y-1) = x^2 /2 + C1 Mr F says: Wrong. It is ln|y - 1| NOT ln(y - 1). There is a big difference that you should be aware of from basic calculus.

Mutliplied both sides by 2

ln(y-1) + x^2 = C Mr F says: If you multiply both sides by 2, and then subtract x^2, then surely you get 2 ln|y - 1| - x^2 = C. Although I don't know why you moved the x^2. It is better left as 2 ln|y - 1| = x^2 + C.

= 1.099 Mr F says: This has come out of nowhere and is meaningless without explanation.

Again do I need to find 'C' to answer this question or just leave it with at the integrated solution?

Thanks
Obviously you need to find the value of C, why do you think you were "given that when x = -, y = 4" ? I assume you are meant to get y as a function of x, so I don't know why you are giving a single number as an answer.

Sorry, but from what I can see your problem is mainly algebra not calculus. You would be well-advised to take much greater care with the algebra. And it wouldn't hurt to review your class notes and textbook on solving differential equations.