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Can anyone provide some techniques for solving this kind of problem? Many thanks.

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2 Answers

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It's easy to see that $\frac{dy}{dx} \rightarrow \infty$ as $|y| \rightarrow \infty$, which narrows the selection down to A and B.

Since $\frac{dy}{dx} > 0$, we would exclude choice B because the derivative of that function is about 0 when its graph intersects the y-axis.

Another way to solve is to set:

$$y' = \frac{dy}{dx}$$

$y'(0) = 1$, so B and E are not an answer. $y'(\infty) = \infty$, so C is not an answer. Further, $y'(-\infty)= \infty$, so D is not an answer either.

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Hint: Since $y^4\geq0$, the equation $y'(x)=1+y^4$ means that the slope of the graph of $y$ is always greater than or equal to 1.

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