Solve the equation. (dy)/(dx)=(x^(2))/(6e^(y)) Choose 1 answer: (A) y=+-sqrt(-2e^(-x)+C)+10 (B) y=+-sqrt(-2e^(-x))+C (C) y=ln((x^(3))/(18)+C) (D) y=ln((x^(3))/(18))+C

Question

Solve the equation.  (dy)/(dx)=(x^(2))/(6e^(y)) Choose 1 answer: (A)  y=+-sqrt(-2e^(-x)+C)+10 (B)  y=+-sqrt(-2e^(-x))+C (C)  y=ln((x^(3))/(18)+C) (D)  y=ln((x^(3))/(18))+C

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Separate variables: We are given the differential equation (dy)/(dx)=(x^(2))/(6e^(y)). This is a separable differential equation, which means we can separate the variables y and x on different sides of the equation. Let's do that.Isolate dy term: Multiply both sides by 6e^(y) to get 6e^(y) * (dy)/(dx) = x^(2). This isolates the dy term on the left side.Divide by x^2: Now, divide both sides by x^(2) to get 6e^(y) * (dy)/(dx) * (1/x^(2)) = 1. This isolates the dx term on the right side.Integrate with respect: We can now integrate both sides with respect to their respective variables. The left side with respect to y and the right side with respect to x. So we have ∫6e^(y) dy = ∫(1/x^(2)) dx.Solve for e^y: The integral of 6e^(y) with respect to y is 6e^(y), and the integral of (1/x^(2)) with respect to x is -1/x. So we have 6e^(y) = -1/x + C, where C is the constant of integration.Take natural logarithm: Now, we divide both sides by 6 to solve for e^(y). This gives us e^(y) = (-1/(6x)) + C/6.Correct constant of integration: To solve for y, we take the natural logarithm (ln) of both sides. This gives us y = ln((-1/(6x)) + C/6).However, there is a mistake in the previous step. The constant of integration should be a separate term, not divided by 6. We should have e^(y) = -1/(6x) + C, and then take the natural logarithm of both sides to get y = ln(-1/(6x) + C).

Solve the equation. $\newline$ $\frac{d y}{d x}=\frac{x^{2}}{6 e^{y}}$ $\newline$ Choose $1$ answer: $\newline$ (A) $y= \pm \sqrt{-2 e^{-x}+C}+10$ $\newline$ (B) $y= \pm \sqrt{-2 e^{-x}}+C$ $\newline$ (C) $y=\ln \left(\frac{x^{3}}{18}+C\right)$ $\newline$ (D) $y=\ln \left(\frac{x^{3}}{18}\right)+C$

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