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Find the solution to the following differential equation:

$$\frac{dy}{dx} = \frac{7x^{13}y}{1+x^{14}}$$

Here are the steps that I took:

$$\frac{1}{y}dy = \frac{7x^{13}}{1+x^{14}}dx$$ $$\int\frac{1}{y}dy = \int\frac{7x^{13}}{1+x^{14}}dx$$ $$\ln y = \frac{1}{2}\ln |1+x^{14}| + C$$ $$\ln y = \ln |(1+x^{14})^{\frac{1}{2}}| + C$$ $$y = C\sqrt{1 + x^{14}}$$

However, this doesn't match any of the answer choices given, so I'm confused which step was incorrect. The choices are (a) $\arcsin( 1 + x^{14})$, (b) $\arccos( 1 + x^{14})$, (c) $\arctan( 1 + x^{14})$, and (d) none of the above. Initially, I was thinking that it could be none of the above but I'm wondering if there are any algebraic manipulations that make it any other answer choice.

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    None of those answers are possible. A quick way you can show this to yourself is to sub in test values. $x = 0$ makes quick work of it. If you want to be sure of your own answer, try putting it back into the question to check. – 2017-02-10
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    ...another test would be to differentiate the "choices" a, b, c. You are correct that the answer is $(d)$ None of the above. – 2017-02-10

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