Solve the given differential equation by using an appropriat

Solve the given differential equation by using an appropriate substitution. The DE is a Bernoulli equation.

dy/dx = y(xy⁵ - 1)

Solution

(dy/dx) = y(xy^5 -1)

dy/dx = xy^6 - y

dy/dx + y = xy^6

divide by y^6

(1/y^6) dy/dx + 1/y^5 = x -----------(1)

let v = 1/y^6
v \' = -(5/y^6) dy/dx

(1/y^6) dy/dx = - (1/5)v \'

substitute in eqn (1)

-(1/5)dv/dx + v = x

dv/dx - 5v = -5x

P(x) = -5 and Q(x) = -5x

Integration factor = e^-5dx = e^(-5x)

v = 1/e^(-5x) [ -5x e^(-5x) dx + C ]

integrate by parts, let u = -5x, du = -5 dx and dv = e^(-5x) dx, v =-(1/5)e^(-5x)

v = e^(5x) [ x e^(-5x) - e^(-5x) dx + C ]

v = e^(5x) [ xe^(-5x) + (1/5)e^(-5x) + C ]

v = x + (1/5) + Ce^(5x)

substitute v = 1/y^5

1/y^5 = 1/5 [ 5x + 1 + Ke^(5x) ]

y^3 = 5 / (5x + 1 + Ke^(5x))

y = [ 5/(5x + 1 + Ke^(5x)) ]^(1/5)