The Bernoulli equation (after Jacob Bernoulli ) is a non-linear ordinary differential equation of the first order of the mold
![y '(x) = f (x) y (x) + g (x) y ^ {\ alpha} (x), \ \ alpha \ notin \ lbrace 0.1 \ rbrace.](https://wikimedia.org/api/rest_v1/media/math/render/svg/c12ccd3d9616daf81850c36c45b9fb23e30fd297)
Through the transformation
![\ z (x): = (y (x)) ^ {{1- \ alpha}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1a470e610027060ed657c85c195c9bf5c262fae)
one can apply it to the linear differential equation
![z '(x) = (1- \ alpha) {\ bigl (} f (x) z (x) + g (x) {\ bigr)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9434bd191f4202e90f58904e28c83822044a9d6d)
lead back.
The equation is not to be confused with the Bernoulli equation of fluid mechanics .
Theorem on the transformation of Bernoulli's differential equation
Be and
![x_ {0} \ in (a, b)](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d03f71523a733bbcdbd4ef0601099717cd1b75)
![\ 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.](https://wikimedia.org/api/rest_v1/media/math/render/svg/0113def11563106d94de7a9a75030e34deb6ac09)
a solution to the linear differential equation
![z '(x) = (1- \ alpha) f (x) z (x) + (1- \ alpha) g (x).](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb492b0ed094d3de3f0b4d2cc5f80413f4dd9dd6)
Then
![y (x): = [z (x)] ^ {{{\ frac {1} {1- \ alpha}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7fda2406fe22a95fb5bf0b126b3ec57e23fc61b)
the solution of Bernoulli's differential equation
![y '(x) = f (x) y (x) + g (x) y ^ {\ alpha} (x) \, \ y (x_ {0}) = y_ {0}: = [z (x_ { 0})] ^ {{{\ frac {1} {1- \ alpha}}}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/90eac927b3996b4b40818534c85f5ad838fce75a)
Furthermore, Bernoulli's differential equation has for each trivially as a solution for .
![\ alpha> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc)
![y \ equiv 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/01c2b2f8ee4ba2f80f9645aa735a5a1929ad9c95)
![y_ {0} = 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/f952fc15fe931e15f2f4a766b3ce68dc52f64842)
proof
It applies
![{\ begin {array} {lll} y '(x) & = & {\ frac {1} {1- \ alpha}} z (x) ^ {{{{\ frac {1} {1- \ alpha}} -1}} z '(x) \\ & = & {\ frac {1} {1- \ alpha}} z (x) ^ {{{\ frac {1} {1- \ alpha}} - 1} } {\ bigl (} (1- \ alpha) f (x) z (x) + (1- \ alpha) g (x) {\ bigr)} \\ & = & f (x) z (x) ^ { {{\ frac {1} {1- \ alpha}}}} + g (x) z (x) ^ {{{\ frac {\ alpha} {1- \ alpha}}}} \\ & = & f ( x) y (x) + g (x) y ^ {\ alpha} (x) \, \ end {array}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d3274e6d0f411f7912f6200ca1f57fb9e312a31)
while the initial value is trivially fulfilled.
![\Box](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Example: Logistic differential equation
The logistic differential equation
![y '(x) = ay (x) -by ^ {2} (x), \ y (0) = y_ {0}> 0, \ a, b> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6836c3685361428a8dd9b852c905030f8273102)
is a Bernoulli differential equation with . So you solve
![\ alpha = 2](https://wikimedia.org/api/rest_v1/media/math/render/svg/938489e6428bb7959330df8c06c79a994811c4a9)
![z '(x) = - az (x) + b \, \ z (0) = {\ frac {1} {y_ {0}}},](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4252ea48f80591545692754e9790ec08a59b628)
surrendered
![z (x) = {\ frac {b} {a}} + \ left ({\ frac {1} {y_ {0}}} - {\ frac {b} {a}} \ right) e ^ {{ -ax}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8e8f3fc6a3cae4baab55e61eabe937d768fd2b3)
As for everyone with
![z (x)> 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/805d5d7421d83d64ab0a5da5bd791d2310c7c5a4)
![x> x ^ {-}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad97511ab7aac1f5d7845afd4fabc7fed1fd1b3c)
![x ^ {-}: = \ left \ {{\ begin {array} {ll} - \ infty \, & {\ textrm {if}} \ a \ geq by_ {0}, \\ {\ frac {1} {a}} \ ln (1 - {\ frac {a} {by_ {0}}}) \, & {\ textrm {if}} \ a <by_ {0}, \\\ end {array}} \ right.](https://wikimedia.org/api/rest_v1/media/math/render/svg/69ada35c6ed0dbd24ffdd9f0b7f7812d87b9119a)
is
![y (x): = {\ frac {1} {z (x)}} = {\ frac {1} {{\ frac {b} {a}} + \ left ({\ frac {1} {y_ { 0}}} - {\ frac {b} {a}} \ right) e ^ {{- ax}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ea68d8a3db1bca1319b63ba579eec59d9c11f21)
the solution of the above equation .
![(x ^ {-}, \ infty)](https://wikimedia.org/api/rest_v1/media/math/render/svg/432f10d018512478090031e472bbdbb222e711f9)
literature
- Harro Heuser: Ordinary differential equations. Teubner, Stuttgart; Leipzig; Wiesbaden 2004, ISBN 3-519-32227-7