Questions: Bernoulli Equations and Substitution Methods

Question 1

Which substitution turns y' + 2y = xy³ into a linear equation?

• A. v = y³
• B. v = y⁻²
• C. v = y²
• D. v = ln y

Answer: (B)

Here n = 3, so v = y^(1-n) = y⁻². Then v' = -2y⁻³y', and dividing the equation by y³ gives y⁻³y' + 2y⁻² = x, which becomes -v'/2 + 2v = x, a linear equation in v.

Question 2

Solve y' + y = y².

Model answer: y = 1/(1 + Ceˣ), together with the solution y = 0.

With n = 2, set v = y⁻¹, so v' = -y⁻²y'. Dividing by y² gives y⁻²y' + y⁻¹ = 1, which becomes -v' + v = 1, or v' - v = -1. Its solution is v = 1 + Ceˣ, so y = 1/(1 + Ceˣ). Dividing by y² excluded y = 0, which also satisfies the equation.

Question 3

Every Bernoulli equation with n ≠ 0, 1 is a linear differential equation in y.

• T. True
• F. False

Answer: False

The term Q(x)yⁿ makes the equation nonlinear in y whenever n is not 0 or 1. The equation becomes linear only in the new variable v = y^(1-n).

Question 4

The logistic equation P' = rP(1 - P/K) can be written P' - rP = -(r/K)P². What is n, and what substitution linearizes it?

• A. n = 1, v = P
• B. n = 2, v = P⁻¹
• C. n = 2, v = P²
• D. n = -1, v = P²

Answer: (B)

The right side is a multiple of P², so n = 2 and the substitution is v = P^(1-2) = 1/P. The resulting linear equation is v' + rv = r/K, whose solution leads to the familiar S-shaped logistic curve.