Part 1) Find the general solution of the differential equation dy/dx + 4xy^6 = 0

Question

Option 1) . y(x) = �(A + 10x^2)^-1/5

Option 2) y(x) = �(A - 12x^2)^-1/6

Option 3) y(x) = �(A + 12x^2)^1/6

Option 4) y(x) = �(A - 10x^2)^-1/5

Option 5) y(x) = �(A + 10x^2)^-1/6

Part 2) Find the particular solution y0 such that y0(0) = 1/2

Option 1) y0(x) = (32 + 12x^2)^1/5

Option 2) y0(x) = (64 - 12x^2)^1/6

Option 3) y0(x) = (64 + 12x^2)^-1/6

Option 4) y0(x) = (32 + 10x^2)^-1/5

Option 5) y0(x) = (32 - 10x^2)^-1/5

Part 3) For the particular solution y0 in (ii), find the value of y0(1)

Option 1) 77^1/6

Option 2) 44^-1/5

Option 3) 76^1/6

Option 4) 43^1/5

Option 5) 42^-1/5

Answer 1

dy/dx + 4xy^6 = 0
dy/dx = -4xy^6
y^-6 dy = -4x dx
-1/5 y^-5 = -2x^2 + C
or
y^5 = 10x^2 + C

Now use y(0) = 1/2 to find C, and
then find y(1)