# Thread: Separable Differential Equation Problem I'm having

1. ## Separable Differential Equation Problem I'm having

Question:
Solve the separable differential equation
Subject to the initial condition: , y = ?

(My work is attached)

Any help would be greatly appreciated!

2. Originally Posted by s3a
Question:

Solve the separable differential equation

Subject to the initial condition: , y = ?

(My work is attached)

Any help would be greatly appreciated!
$3x - 4y\sqrt{x^2 + 1}\,\frac{dy}{dx} = 0$

$\frac{3x}{\sqrt{x^2 + 1}} - 4y\,\frac{dy}{dx} = 0$

$\frac{3x}{\sqrt{x^2 + 1}} = 4y\,\frac{dy}{dx}$

$\int{\frac{3x}{\sqrt{x^2 + 1}}\,dx} = \int{4y\,\frac{dy}{dx}\,dx}$

$\int{3x(x^2 + 1)^{-\frac{1}{2}}\,dx} = \int{4y\,dy}$.

You should be able to evaluate both integrals now. To do the left hand side, you need to use the substitution $u = x^2 + 1$ so that $\frac{du}{dx} = 2x$.

3. Did you check my work? (because that's exactly what I did)

Originally Posted by Prove It
$3x - 4y\sqrt{x^2 + 1}\,\frac{dy}{dx} = 0$

$\frac{3x}{\sqrt{x^2 + 1}} - 4y\,\frac{dy}{dx} = 0$

$\frac{3x}{\sqrt{x^2 + 1}} = 4y\,\frac{dy}{dx}$

$\int{\frac{3x}{\sqrt{x^2 + 1}}\,dx} = \int{4y\,\frac{dy}{dx}\,dx}$

$\int{3x(x^2 + 1)^{-\frac{1}{2}}\,dx} = \int{4y\,dy}$.

You should be able to evaluate both integrals now. To do the left hand side, you need to use the substitution $u = x^2 + 1$ so that $\frac{du}{dx} = 2x$.

4. 1. Don't be so rude - most people on this forum do not open attachments.

2. When you have got it to the step:

$\frac{3}{2}\int{u^{-\frac{1}{2}}\,du} = \int{4y\,dy}$

$3u^{\frac{1}{2}} + C_1 = 2y^2 + C_2$

$3\sqrt{x^2 + 1} + C_1 - C_2 = 2y^2$

$\frac{3}{2}\sqrt{x^2 + 1} + C = y^2$, where $C = \frac{1}{2}(C_1 - C_2)$

$y = \pm \sqrt{\frac{3}{2}\sqrt{x^2 + 1} + C}$.

If $y(0) = 1$

$1 = \pm \sqrt{\frac{3}{2}\sqrt{0^2 + 1} + C}$

$1 = \frac{3}{2}\sqrt{1} + C$

$1 = \frac{3}{2} + C$

$C = -\frac{1}{2}$.

So $y = \pm \sqrt{\frac{3}{2}\sqrt{x^2 + 1} - \frac{1}{2}}$

$y = \pm \sqrt{\frac{3\sqrt{x^2 + 1} - 1}{2}}$

$y = \pm \frac{\sqrt{3\sqrt{x^2 + 1} - 1}}{\sqrt{2}}$

$y = \pm \frac{\sqrt{6\sqrt{x^2 + 1} - 2}}{2}$.

5. I get it now thanks and I didn't try to sound rude, it just probably came that way to you because it's hard to see "emotion" (for the lack of a better word) through text.