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

1 Answer
Aug 22, 2016

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

Explanation:

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