What is a solution to the differential equation #dy/dx=xy^2# with the particular solution #y(2)=-2/5#?

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Answer 1 · 2016-07-11T21:49:51

# y = - 2/(x^2 + 1)#

#dy/dx=xy^2#

this is separable

#1/y^2 \ dy/dx=x#

#int \ 1/y^2 \ dy/dx \ dx =int \ x \ dx#

#int \ 1/y^2 \ dy =int \ x \ dx#

# - 1/y = x^2/2 + C#

#y(2) = - 2/5#

# 5/2 = 2 + C implies C = 1/2#

# - 1/y = 1/2 (x^2 + 1)#

# - 2/y = (x^2 + 1)#

# - y/2 = 1/(x^2 + 1)#

# y = - 2/(x^2 + 1)#