What is the general solution of the differential equation? : # y' = x(1+y^2) #

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Answer 1 · 2018-02-07T18:00:49

# y = tan(1/2x^2 + C) #

We have:

# y' = x(1+y^2) #

This is a First Order Separable Ordinary Differential Equation. We can rewrite in the form:

# 1/(1+y^2) dy/dx = x #

So we can "separate the variables" to get:

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

Both integrals are standard calculus results , so integrating we get:

# arctany = 1/2x^2 + C #

Leading to the General Solution:

# y = tan(1/2x^2 + C) #