Antiderivatives Calculus

Question

Find the particular antiderivative that satisfies the following conditions:

dy/dx = (4x+2)/ (x^1/3); y(1) = 5

y =

Answer 1

Given differential equation:

dy/dx = (4x+2)/ (x^1/3); y(1) = 5

By performing the division, we should obtain the following:

dy/dx = 4x^2/3 + 2x^-1/3

Then integrate both sides of the equation:

∫dy = ∫(4x^2/3 + 2x^-1/3) dx

y = 4(3/5) x^5/3 + 2(3/2)x^2/3 + C

y = (12/5)x^5/3 + 3 x^2/3 + C

y(1)=5 means if x=1, y=5. Plug it in to the equation:

5 = (12/5)(1)^5/3 + 3 (1)^2/3 + C

5 = 12/5 + 15/5 + C

5 = 27/5 + C

25/5 - 27/5 = C

C = -2/5

The particular solution for this differential equation is:

y = (12/5)x^5/3 + 3 x^2/3 - 2/5