A differential equation of Bernoulli type is written as

This type of equation is solved via a substitution. Indeed, let

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

Note that if n > 1, then we have to add the solution y=0 to the solutions found via the technique described above.

(1) Recognize that the differential equation is a Bernoulli equation. Then find the parameter n from the equation;

(2) Write out the substitution

(3) Through easy differentiation, find the new equation satisfied by the new variable v.

You may want to remember the form of the new equation:

(4) Solve the new linear equation to find v;

(5) Go back to the old function y through the substitution

(6) If n > 1, add the solution y=0 to the ones you obtained in (4).

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




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