What is a solution to the differential equation dydx=xy2 with the particular solution y(2)=−25?

dydx=xy2this is separable1y2dydx=x∫1y2dydxdx=∫xdx∫1y2dy=∫xdxpower rule−1y=x22+Cy(2)=−2552=2+C⇒C=12−1y=12(x2+1)−2y=(x2+1)−y2=1x2+1y=−2x2+1

"What is a solution to the differential equation..." solved and explained

fernandoval3sbr

2022-04-19

What is a solution to the differential equation

\frac{d y}{d x} = x y^{2}

with the particular solution

y (2) = - \frac{2}{5}

wyjadaczeqa8

Beginner2022-04-20Added 14 answers

\frac{d y}{d x} = x y^{2}

this is separable

\frac{1}{y^{2}} \frac{d y}{d x} = x

\int \frac{1}{y^{2}} \frac{d y}{d x} d x = \int x d x

\int \frac{1}{y^{2}} d y = \int x d x

power rule

- \frac{1}{y} = \frac{x^{2}}{2} + C

y (2) = - \frac{2}{5}

\frac{5}{2} = 2 + C \Rightarrow C = \frac{1}{2}

- \frac{1}{y} = \frac{1}{2} (x^{2} + 1)

- \frac{2}{y} = (x^{2} + 1)

- \frac{y}{2} = \frac{1}{x^{2} + 1}

y = - \frac{2}{x^{2} + 1}

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