What is a solution to the differential equation #dy/dx=x^2(1+y)# with y=3 when x=0?

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Answer 1 · 2016-08-22T17:08:04

#y = 4e^ (x^3/3) - 1#

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

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

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

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

#ln (1+y) = x^3/3 + C#

#1+y = e^ (x^3/3 + C) = e^ (x^3/3)e^C = Ce^ (x^3/3) #

#y = Ce^ (x^3/3) - 1#

Applying the IV:

#3 = C e^0 -1 = C - 1 implies C = 4#

#y = 4e^ (x^3/3) - 1#