<s>[INST] Differential Equation Help? Q: Solve the separable differential equation:
$3x - 2y\sqrt{x^2+1}\frac{dy}{dx} = 0$ and subject to the initial condition: $y(0) = 4$. What does $y$ equal, in relation to $x$? I am just really confused what I am suppose to do? Thanks.
 [/INST] So what we want here is a particular solution to our ODE given our initial condition: $y(0) = 4$
$$3x\ dx = 2y\sqrt{x^{2}+1}\ dy~;\hspace{20 pt} y(0)=4$$
$$\Rightarrow \displaystyle\int 2y\ dy = \displaystyle\int \dfrac{3x}{\sqrt{x^{2}+1}}\ dx$$
$u = x^{2}+1$
$du = 2x\ dx$
$dx = \dfrac{1}{2x}du$
Making the substitutions in for u and dx we see we come to the following:
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow y^{2} = 3 \cdot \dfrac{1}{2} \displaystyle\int \dfrac{x}{\sqrt{u}} \cdot \dfrac{1}{x}\ du$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow y^{2} = \dfrac{3}{2} \displaystyle\int \dfrac{1}{\sqrt{u}}\ du$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow y^{2} = \dfrac{3}{2} \displaystyle\int u^{-\tfrac{1}{2}}\ du$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow y^{2} = \dfrac{3}{2} \cdot 2 ~ u^{\tfrac{1}{2}} + C$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow y^{2} = 3 \sqrt{u}\ du$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow y^{2} = 3 \cdot \sqrt{x^{2}+1} + C$
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\Rightarrow y(x) = \displaystyle\pm \sqrt{3} \sqrt[4]{x^{2}+1} + C$
$\underline{\text{Applying Initial Conditions:}}$
$y(0)=4:~~4=\sqrt{3} \displaystyle\sqrt[4]{(0)^{2}+1}~ + C$
$~~~~~~~~~~~~~~~~~~~4=\sqrt{3}\cdot 1~ + C$
$~~~~~~~~~~~~~~~~~~~C=4-\sqrt{3}~\doteq~2.3$
$y(0)=4:~~4=-\sqrt{3} \displaystyle\sqrt[4]{(0)^{2}+1}~ + C$
$~~~~~~~~~~~~~~~~~~~4=-\sqrt{3}\cdot 1~ + C$
$~~~~~~~~~~~~~~~~~~~C=4+\sqrt{3}~\doteq~5.7$
Thus the particular solutions to this separable ODE is,
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\therefore~~y(x) = \displaystyle \sqrt{3} \sqrt[4]{x^{2}+1} + 2.3 ~~~~\&~~~~ y(x) = \displaystyle -\sqrt{3} \sqrt[4]{x^{2}+1} + 5.7~~~~\blacksquare$
I hope this helped out, and hopefully I did not make any mistakes to cause any type of confusion. 
Let me know if there is any step that did not make since to you and I will try and clarify a some more.
Thanks.
Good Luck.
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