Which of the following transformations reduce the differential equation d z d x + z x z= z x^2 ( z)^2 into the form d u d x +P(x) u=Q(x)

Step-by-step solution

Given the differential equation dz dx + z x z = z x^2 ( z)^2 . Divide the entire equation by z to obtain 1 z dz dx + z x = ( z)^2 x^2 . Let v = z . Then dv dx = 1 z dz dx . Substituting this into the equation gives dv dx + v x = v^2 x^2 . This is a Bernoulli differential equation of the form dv dx + P(x)v = Q(x)v^n where n=2 . To linearize this, divide by v^2 : v⁻² dv dx + v⁻¹ x = 1 x^2 . Let u = v⁻¹ = ( z)⁻¹ . Then du dx = -v⁻² dv dx , which implies v⁻² dv dx = - du dx . Substituting into the equation: - du dx + u