Solve the differential equation. dy/dx = xy^2

Accepted Solution

You can solve this differential equation by separating the variables and differentiating both sides.

1) Multiply both sides by dx and divide both sides by [tex] y^{2} [/tex].
[tex] \frac{dy}{dx} = xy^2\\ \frac{1}{y^2} \: dy = x\:dx [/tex]

2) Integrate both sides. Remember the power rule for integrals. Say you have a value [tex] x^{n} [/tex], where [tex]n \neq -1[/tex]. Take the power, n, and add 1. Then divide the new expression [tex] x^{n+1} [/tex] by the new power, n + 1. The integral of [tex] x^{n} [/tex] would be [tex] \frac{x^{n+1}}{n+1} [/tex] (+C, if it is an indefinite integral). Remember that you can subtract C from both sides and just have C on one side (since the constant doesn't have a definite value):
[tex]\frac{1}{y^2} \: dy = x\:dx\\ \int \frac{1}{y^2} \: dy= \int x\:dx\\ \int y^{-2} \: dy= \int x\:dx \\ \frac{ y^{-2+1}}{-2+1} + C = \frac{x^{1+1}}{1+1} + C\\ - y^{-1} = \frac{ x^{2}}{2} + C\\ -\frac{2}{y} = x^{2} + C\\ y = - \frac{2}{x^{2} + C} [/tex]

Your solution is [tex]y = - \frac{2}{x^{2} + C} [/tex].