Meeting Mr. Bernoulli

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The differential equation

{d \over dx} y + f(x) y = g(x) y^{\alpha}\quad\quad\quad(1)

where \alpha \neq 0, 1 and g(x) \not \equiv 0, is known as the Bernoulli’s equation.

When \alpha is an integer, (1) has trivial solution y(x) \equiv 0.

To obtain nontrivial solution, we divide each term of (1) by y^{\alpha} to get,

\boxed{y^{-\alpha}{d \over dx}y} + f(x) y^{1-\alpha} = g(x)\quad\quad\quad(2)

Since  {d \over {dx}}({{1 \over {1-\alpha}}y^{1-\alpha}}) ={1 \over {1-\alpha}}\cdot (1-\alpha) y^{1-\alpha-1}{d \over dx}y=\boxed{y^{-\alpha}{d \over dx}y}

(2) can be expressed as

{d \over dx} ({{1 \over {1-\alpha}} y^{1-\alpha}}) + f(x) y^{1-\alpha} = g(x)

which is

{{1 \over {1-\alpha}} {d \over dx} y^{1-\alpha}} + f(x) y^{1-\alpha} = g(x) .

Multiply 1-\alpha throughout,

{d \over dx} y^{1-\alpha} + (1-\alpha) f(x) y^{1-\alpha} = (1-\alpha) g(x)\quad\quad\quad(3)

Let z = y^{1-\alpha}, (3) is transformed to a first order linear equation

{d \over dx} z + (1-\alpha) f(x) z = (1-\alpha) g(x),

giving the general solution of a Bernoulli’s equation (see Fig. 1)

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Fig. 1

For a concrete example of Bernoulli’s equation, see “What moves fast, will slow down

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2 thoughts on “Meeting Mr. Bernoulli

  1. Pingback: What goes up fast, will slow down, Part One | Vroom

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