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Problem 14 Medium Difficulty

Find the solution of the differential equation that satisfies the given initial condition.
$ x + 3y^2 \sqrt {x^2 + 1} \frac {dy}{dx} = 0, y(0) = 1 $

Answer

$y=\left(-\sqrt{x^{2}+1}+2\right)^{1 / 3}$

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Video Transcript

this question asks us Determine the solution of the differential equation. What we know is that we're going to be separating everything with white on one side and then everything with X to the other side. So given expose three y squared square root of X squared plus one, do you? Why over DX is zero. We know we have three. Why squared, do you? Why? Because remember everything with wide to the left hand side everything with extra right hand side. Now integrate both sides and we end up with why Cubed is negative square root of X squared plus one and then do not forget the constant. See, this is very important to the constant integration because now we have the initial value Wives, zeros one. I'm going to be using that to find seat. So if why is one we're plugging? Why? And for one and then X zeros Replacing an extra zero see has to be to therefore plug into the integrated equation. We figured out what aren't you see us? There should not be the sea in the final answer and then to the power of 1/3 is the same as the cube drew essentially