quinta-feira, 3 de outubro de 2013

Bernoulli differential equation

y'+ P(x)y = Q(x)y^n\,
is called a Bernoulli equation when n≠1, 0, which is named after Jacob Bernoulli, who discussed it in 1695 (Bernoulli 1695). Bernoulli equations are special because they are nonlinear differential equations with known exact solutions.

Solution[edit source]

Let x_0 \in (a, b) and
\left\{\begin{array}{ll} z: (a,b) \rightarrow (0, \infty)\ ,&\textrm{if}\ \alpha\in \mathbb{R}\setminus\{1,2\},\\ z: (a,b) \rightarrow \mathbb{R}\setminus\{0\}\ ,&\textrm{if}\ \alpha = 2,\\\end{array}\right.
by a solution of the linear differential equation
z'(x)=(1-\alpha)P(x)z(x) + (1-\alpha)Q(x).
Then we have that y(x) := [z(x)]^{\frac{1}{1-\alpha}} is a solution of
y'(x) = P(x)y(x) + Q(x)y^\alpha(x)\ ,\ y(x_0) = y_0 := [z(x_0)]^{\frac{1}{1-\alpha}}.
And for every such differential equation, for all \alpha>0 we have y\equiv 0 as solution for y_0=0.

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