Bernoulli Equations

A differential equation of Bernoulli type is written as

This type of equation is solved via a substitution. Indeed, let v = y^1-n. Then easy calculations give

which implies

This is a linear equation satisfied by the new variable v. Once it is solved, you will obtain the function y = v^1/(1-n) . Note that if n > 1, then we have to add the solution y = 0 to the solutions found via the technique described above.

Let us summarize the steps to follow:

Recognize that the differential equation is a Bernoulli equation. Then find the parameter n from the equation. Write out the substitution v = y^1-n.

Through easy differentiation, find the new equation satisfied by the new variable v. You may want to remember the form of the new equation:

Solve the new linear equation to find v, then go back to the old function y through the substitution y = v^1/(1-n).

If n > 1, add the solution y = 0 to the ones you obtained.

If you have an IVP, use the initial condition to find the particular solution.

Example

Solve the following equation using reduction to Linear Form. First write the expression as a Bernoulli equation and solve the resultant relationship as a first order linear equation:

Solution: Rewrite base equation as

or

Recognize the above equation as a specific form of Bernoulli’s equation,

y' + p(x)y = g(x)y^a,

where a = -1. Therefore, let u(x) = y^1-a = y². Then the resultant linear equation becomes u' + (1 - a)pu = (1-a)g, or, for this case, and this is a linear first order differential equation.

Now solve the linear first order ODE using the integrating factor approach , where the integrating factor, g(x), is given by (note that this g(x) is different from above)

Therefore,

and, integrating both sides, gives

or

Substitute u = y² to get the final expression