Question

Use the substitution v=y¹⁻ⁿ to transform the following Bernoulli ferential equation into an first order linear differential equation dy/dx - y= eˣy²

Answer 1

To transform the Bernoulli equation dy/dx - y = exy2 into a linear equation, we use the substitution v = y-1, derive dv/dx and substitute back into the equation, resulting in dv/dx + v = ex, a first order linear differential equation.

Step-by-step explanation:

To transform the given Bernoulli differential equation dy/dx - y = exy2 using the substitution v=y1-n, where n=2, we substitute v = y-1 (since 1 - n = -1). This substitution will turn the Bernoulli equation into a first order linear differential equation.

First, we find the derivative of v with respect to x using the chain rule: dv/dx = dv/dy × dy/dx. Since v=y-1, then dv/dy = -y-2 or -1/v2. Multiply this by dy/dx to get dv/dx.

Substituting v and dv/dx back into the original equation gives us: dv/dx + v = ex. This is now a linear equation in standard form dv/dx + P(x)v = Q(x), where P(x) = 1 and Q(x) = ex.