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Solve the differential equation (x + y3) + 6xy2y′ = 0

Use a change of variables to convert the following differential equation into a linear differential equation, and then solve the equation: Incomplete.

1. tom says:

bernoulli :)

• Evangelos says:

Yes indeed, Mr. Tom. Now, for those of you who don’t get the reference, earlier in this chapter, we were introduced to a form of the linear differential equation called the “Bernoulli Equation”, where we could transform a nonlinear first order equation into a linear first order equation for a new, unknown function. For further reading, see section 8.5 exercise #13, RoRi posted the solution here:

http://stumblingrobot.com/2016/01/30/find-all-solutions-of-a-given-initial-value-problem/

Now, onto our current exercise. We have the implicit first order equation Which we can re-write as follows And following from the proof in exercise 8.5 #13, we define a new function v such that: The equation becomes Which is a linear first order equation. From Theorem 8.3, the solution to the above equation, v = g(x), with g(a) = b is as follows: With Giving us: For x>0, as presented in the back of the book.

Now, for a solution for all x, we take the linear equation in v And we see that the equation can be satisfied by a first degree polynomial   