solve the separable differential equation. (use c for any needed constant.)\n\\( \\frac { d y } { d x } = 5…

Explanation:

Step1: Separate variables

Separate the variables in the differential equation (\frac{dy}{dx}=5xy^{2}). We get (\frac{dy}{y^{2}} = 5x dx) (since (y\neq0)).

Step2: Integrate both sides

Integrate (\int y^{- 2}dy=\int5x dx). Using the power - rule for integration (\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C) ((n\neq - 1)), we have: For the left - hand side, (\int y^{-2}dy=\frac{y^{-2 + 1}}{-2+1}+C_1=-\frac{1}{y}+C_1). For the right - hand side, (\int5x dx=5\times\frac{x^{2}}{2}+C_2=\frac{5}{2}x^{2}+C_2). So, (-\frac{1}{y}=\frac{5}{2}x^{2}+C) (where (C = C_2 - C_1)).

Step3: Solve for (y)

First, rewrite the equation as (\frac{1}{y}=-\frac{5}{2}x^{2}-C). Then (y=\frac{1}{- \frac{5}{2}x^{2}-C}), or (y =-\frac{2}{5x^{2}+2C}). Let (K=-2C), then (y=\frac{-2}{5x^{2}+K})

Answer:

(y=\frac{-2}{5x^{2}+C})