Find the particular solution to the differential equation dy/dx=x^2+1/3y when f(3)=1

Answer 1

3y dy = (x^2+1) dx

(3/2) y^2 = (1/3)x^3 + x + c

f(3)=1

(3/2)(1^2) = (1/3)(27) + 3 + c

3/2 = 24/2+ c

3/2 = 24/2 + c

c = -21/2

(3/2) y^2 = (1/3)x^3 + x - 21/2

y^2 = (2/3) [ (1/3)x^3 + x - 21/2]

= (2/9) x^3 + (2/3)x - 7

y = sqrt[ (2/3)x^3 + (2/3)x - 7 ]