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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.

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    @Pacciu: I was hoping you'd read this! (He's the one who first showed me how *primitive* I was being)2011-03-31

2 Answers 2

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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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Here's what you are supposed to do

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

You have a given initial condition and an undetermined constant..