Write and then solve for y = f(x) the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to y4

Question

Write and then solve for y = f(x) the differential equation for the statement: "The rate of change of y with respect to x is inversely proportional to y^4."

Answer 1

dy/dx = k/y^4

y^4 dy = k dx

integrating

(1/5)y^5 = kx+C

y = 5(kx+C)^(1/5)

check:

differentiating:

dy/dx = 5(k/5)(kx+C)^(-4/5)

5(k/5)[(kx+C)^(1/5)]^(-4)

= 5(k/5) ( y)^(-4)

= 5(k/5) (y ^(-4))

= (k/y^4)

Answer is official:

y = 5(kx+C)^(1/5)

Answer 2

dy/dx = k/y⁴

y⁴dy = kdx

1/5 y⁵ = kx + C

y = ⁵√(k₁x + C₁)