Question

Find the solution of the differential equation dy/dx=lnx/(xy) that satisfies the initial condition y(1)=−3

Answer 1

We can separate the variable quantities to get\[ydy =\frac{ \ln x }{ x }dx\]do you see that? If yes, then what do you think should be done next.

Answer 2

@manon1992 ??

Answer 3

Integrate both parts?

Answer 4

Correct!

Answer 5

Okay, I had not seen that thank you :)

Answer 6

I got it thanks a lot!! :)

Answer 7

Okay! Why don't you finish it an show me your work so that I can check if you've done them correctly.

Answer 8

Oh okay. Good Luck!

Answer 9

thank you!

Answer 10

actually no I don't have it haha
in the end we get y²=ln²x+9 right?

Answer 11

oh no that's another exercice. I took y(-3)=1

Answer 12

So we have\[\int\limits_{}^{}ydy =\int\limits_{}^{}\frac{ \ln x }{ x }dx\]First what do we get on each side?

Answer 13

1/2y²=&/2ln²x+c ?

Answer 14

I want to solve it with y(1)=-4

Answer 15

Right! We get\[\frac{ 1 }{ 2 }y ^{2}=\frac{ 1 }{ 2 }\ln ^{2}x +c\]Now is the initial condition y(1) = -3 or y(1) = -4 ??? It seems you keep changing your mind lol. Regardless of which it is, do you know what to do to find the constant c?

Answer 16

it is y(1)=-4 sorry
I would replace y² on the RHS by -4 and we would get : 1/2(-4)²=1/2ln²(1)+c
Knowing that ln1=0, c=1/2(-4)²=8?

Then we would have 1/2y²=1/ln²x+8 ?

Answer 17

Perfect!!! Good Job!

Answer 18

then, y=(ln²x+16)^1/2 ?

Answer 19

it doesn't work...

Answer 20

What do you mean?

Answer 21

Do you mean the answer is given differently?

Answer 22

It is an exercise I have to do on webwork and I can see when it's wrong or right and they don't accept this answer

Answer 23

OK, well what if you leave it with y^2

Answer 24

I can't there is y= and then a blank I have to fill...

Answer 25

Because that is right, based on the equation and the initial condition you provided.

Answer 26

I know, I will figure it out :) thank you!

Answer 27

It could be because they want both the positive and the negative root. Hence +/-

Answer 28

It appears not... I attached a screen shot

Answer 29

ur initial condition = -4 so y=lnx -4

Answer 30

ya

Answer 31

If you look at there answer, it contains ln(x)². Suggesting that the x is squared, not the logarithm function. But clearly it should be the function that is squared. In other words, it should be (ln (x))². Therefore the final answer should be y = (ln(x))² + 16)^(1/2) = (ln²(x) + 16)^(1/2). So clearly they have made a major error!

Answer 32

oh u are right i made a mistake with the constant

Answer 33

still isn't working :/ but thank you