Bernoulli

Differential Equations

The Bernoulli Differential Equation has an analytical solution which can be found two separate ways

Bernoulli Method

The Bernoulli equation $y^{'} + p (x) y = g (x) y^{α}$ can be reduced to two separable equations. We seek its solution as the product of two functions $y (x) = u (x) v (x)$ , where $u (x)$ is a solution of a linear part.

u^{'} + p (x) u = 0 ⟹ d u = - p (x) d x

Upon integration we find:

u (x) = exp (- \int p (x) d x)

The second multiple, $v (x)$ , is the general solution of the "truncated" Bernoulli equation without a linear part:

v^{'} = g (x) u^{α - 1} v^{α} ⟹ d u = g (x) u^{α - 1} d x

The separable equation is also integrated without a problem:

v^{1 - α} = (1 - α) \int g (x) u^{α - 1} d x

Taking the $1 - α$ root and multiplying the result by $u (x)$ yields the required solution.

y (x) = exp (- \int p (x) d x) [(1 - α) \int g (x) u^{α - 1} d x]^{1/ (1 - α)}