Question

Solve the separable differential equation 3x-4ysqrt(x^2+1)dy/dx=0 Subject to the initial condition: y(0)=-9

Answer 1

What have you tried with this problem so far? Do you know how to start? :)

Answer 2

I have tried moving the 4ysqrt(x^2+1)dy/dx to the other side and then dividing both sides by \[\sqrt{x^2+1}\] and the separating the variables and bring dx over with the other x's

Answer 3

If you have it separated, you integrate from your lower bound on x to x and same for the y side.

Answer 4

Okay, that sounds correct so far. :)

So we'd have something like this:
\( \displaystyle 4y \; \text{d}y = \frac{3x}{\sqrt{x^2 + 1}} \; \text{d} x \)
Aye, then we just integrate now as it is separated.

Answer 5

After I integrated both sides I get 3sqrt(x^2+1+C= 2y^2 but when I plug in y(0) = -9 I find C to be 159 I then solve for y and get y = \[\sqrt{(3\sqrt{x^2+1}+159)/2}\]

Answer 6

But when I enter it it says it is wrong

Answer 7

Youd want to do
\[\int_{-9}^{y}4t\ dt = \int_{0}^{x}\frac{3s}{\sqrt{s^{2}+1}}ds\] then solve for y

Answer 8

After I solve for y I should get y = \[\sqrt{\frac{ 3\sqrt{x^2+1}+159 }{ 2 }}\] right?

Answer 9

Thanks

Answer 10

Sorry, I just thought about it, but y(0) = -9 means that we have the negative root rather than the positive. We could have either positive or negative initially, but the only way it can be negative is if its all negative (the sqrt will not create negatives).