Solve a Bernoulli Equation

Question

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

x(dy/dx)+y=1/(y^2)

Answer 1

Bernoulli's Equation follows the form dy/dx + yP(x) = (yⁿ)Q(x).

Here, x(dy/dx) + y = 1/y² can go to dy/dx + y(1/x) = (y^-2)(1/x), which

obeys the form above.

Make the transformation v = y^-(-2)+1 or y³ and y = v^1/3.

Then dy/dx = (1/3)v^-2/3(dv/dx).

Rewrite dy/dx + y(1/x) = (y^-2)(1/x) as (1/3)v^-2/3(dv/dx) + v^1/3(1/x) = (v^1/3)^-2(1/x).

Multiply through this last by 3v^2/3 to gain dv/dx + 3v(1/x) = 3/x.

Note that dv/dx + 3v(1/x) = 3(1/x) is a linear differential equation of the first order with

integrating factor e^∫(3/x)dx.

Then build the general solution or "primitive" as ve^3ln|x| equals ∫(3/x) • e^3ln|x|dx.

Rewrite this last as y³(x³) = ∫[(3/x) • x³]dx or x³y³ = x³ + C or y³ = 1 + Cx^-3.

Finally, express y³ = 1 + Cx^-3 as y = (1 + Cx^-3)^1/3.