Question

(a) Solve Bernoulli's equation when n = 0; when n = 1.

(b) Show that if n≠ 0, 1, then the substitution = y¹-^n reduces Bernoulli's equation to a linear equation. This method of solution was found by Leibniz in 1696.

Answer 1

Bernoulli's equation can be solved when n=0 and n=1. When n≠ 0, 1, the substitution y = u^(1-n) reduces Bernoulli's equation to a linear equation. This method was found by Leibniz in 1696.

Step-by-step explanation:

Bernoulli's equation is a mathematical equation that relates pressure, velocity, and height in a fluid. When n=0, Bernoulli's equation simplifies to P1 + pgh1 = P2, where P1 and P2 are the pressures at two different points in the fluid and h1 is the height at the first point. When n=1, Bernoulli's equation becomes P1 + pgh1 + 0.5pV1^2 = P2 + pgh2 + 0.5pV2^2, where V1 and V2 are the velocities at the two points.

To show that if n≠ 0, 1, the substitution y = u^(1-n) reduces Bernoulli's equation to a linear equation, we substitute the expressions for pressure and velocity into Bernoulli's equation and simplify. This results in a linear equation in terms of y.

This method of solution using the substitution y = u^(1-n) was found by Leibniz in 1696.