Bernoulli Equations

Bernoulli equations are of the form:

\[\frac{dy}{dx}+P(x)y=Q(x)y^n\]

Dividing by \(y^n\):

\[\frac{dy}{dx}y^{-n}+P(x)y^{1-n}=Q(x)\]

If \(n=0\) or \(n=1\) then the equation is linear.  Otherwise a substitution of \(u=y^{1-n}\) will give us a linear equation.

If \(u=y^{1-n}\), then \(\frac{du}{dx}u=(1-n)\frac{dy}{dx}y^{-n}\).  Substituting these into the second equation from above:

\[\frac{1}{1-n}\frac{du}{dx}+P(x)u=Q(x)\]

For example,

\[y'+2xy=xy^4\]

Dividing by \(y^4\), we get  \(y'y^{-4}+2xy^{-3}=x\)

Substituting \(u=y^{-3}\) and \(\frac{du}{dx}={-3}\frac{dy}{dx}y^{-4}\) gives us

\[\frac{1}{-3}\frac{du}{dx}+2xu=x\]

\[\frac{du}{dx}-6xu=-3x\]

Leave a Response

You must be logged in to post a comment.