solve the separable differential equation. (use c for any needed constant.)\n\\frac{dy}{dx}=3xy^{2}\\…

solve the separable differential equation. (use c for any needed constant.)\n\\frac{dy}{dx}=3xy^{2}\\ (y\\neq0)

solve the separable differential equation. (use c for any needed constant.)\n\\frac{dy}{dx}=3xy^{2}\\ (y\\neq0)

Answer

Explanation:

Step1: Separate variables

Separate (y) and (x) terms: (\frac{dy}{y^{2}}=3x dx) ((y\neq0)).

Step2: Integrate both sides

Integrate (\int y^{- 2}dy=\int3x dx). Using the power rule (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), we have (\frac{y^{-2 + 1}}{-2+1}=\frac{3x^{2}}{2}+C). Simplify to (-\frac{1}{y}=\frac{3x^{2}}{2}+C).

Step3: Solve for (y)

First, rewrite the equation as (\frac{1}{y}=-\frac{3x^{2}}{2}-C). Then (y = \frac{1}{-\frac{3x^{2}}{2}-C}=\frac{-2}{3x^{2}+2C}). Let (C_{1}=- 2C), so (y=\frac{-2}{3x^{2}+C_{1}}).

Answer:

(y=\frac{-2}{3x^{2}+C})