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 y4."

Answer 1

dy/dx = k/y⁴

y⁴dy = kdx

1/5y⁵ = kx + C

y⁵ = k₁x + C₁

y = ⁵√(k₁x + C₁). (The subscripts just convey that these 2 are different constants than those in the first equation written after integrating.)

Answer 2

Rate of change of y with respect to x is dy/dx. Inversely proportional means that rate = k/y⁴ so:

dy/dx = k/y⁴

cross multiplying gives us y⁴ dy = k dx now integrate both sides:

∫y⁴ dy = ∫k dx

y⁵/5 = kx + C or

y⁵ = C₁x + C₂

y = ⁵√(C₁x + C₂)