First-order equations · Definition
Bernoulli equation
Before you start you need to know
An almost linear equation, with a power of \(y\) in the way, is rescued by a change of variable that returns it to linear form.
Definition
A Bernoulli equation is a first-order linear equation whose independent term \(q\) is multiplied by a power \(y^n\) of the unknown, with \(n\ne0,1\).
If \(n=0\) or \(n=1\) the equation is already linear, which is why they are excluded. For any other \(n\) the term \(y^n\) makes it non-linear and the integrating factor does not apply directly. Jakob Bernoulli studied this type in the 17th century, and the substitution that solves it is due to Leibniz.
The trick is to divide by \(y^n\) and set \(u=y^{1-n}\). Since \(u'=(1-n)y^{-n}y'\), the equation becomes \(u'+(1-n)p\,u=(1-n)q\), which is linear in \(u\). Solve it with the integrating factor and return with \(y=u^{1/(1-n)}\).
Guiding example: in \(y'+y=y^2\) take \(u=1/y\), so \(u'-u=-1\) and \(u=1+Ce^{x}\); then \(y=\frac{1}{1+Ce^{x}}\). It is a logistic model in disguise: the solution tends to \(0\) when \(C>0\) and \(x\to\infty\).
Watch two things: dividing by \(y^n\) (for \(n>0\)) loses the solution \(y=0\), which must be added separately; and in the final linear equation the coefficient carries the factor \((1-n)\), whose sign is a source of errors. Later, autonomous equations and the logistic model revisit this kind of result.
Think about it
Which substitution would you use in \(y'+y=y^{3}\), and what linear equation results?
\(u=y^{-2}\), giving \(u'-2u=-2\).
Exercises 10
- Which of these equations is a (non-linear) Bernoulli equation?
- For \(y'+y=xy^2\), which substitution makes it linear?
- With \(u=1/y\), the equation \(y'+y=xy^2\) becomes:
- In \(y'+p(x)\,y=q(x)\,y^n\), for which \(n\) is the equation already linear?
- Solve \(y'+y=y^2\) (with \(y\ne0\)).
- Solve \(y'+\dfrac{y}{x}=xy^2\) for \(x>0\).
- If \(y'+y=y^2\) and \(y(0)=\tfrac12\), what is \(y(\ln 3)\)?
- With \(u=y^{-2}\), the equation \(y'+y=y^3\) becomes:
- For \(n>0\), when dividing \(y'+py=qy^n\) by \(y^n\), which solution can be lost?
- If \(y'=y+y^3\), what is \(y^2\) in terms of \(C\)?
References
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade, Elementary Differential Equations and Boundary Value Problems, 12th edition, Wiley, 2021. §2.4
- Dennis G. Zill, A First Course in Differential Equations with Modeling Applications, 11th edition, Cengage, 2018. §2.5