Bernoulli Equations

Bernoulli equations are of the form:


Dividing by \(y^n\):


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:


For example,


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



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