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

**Incomplete.**

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Stumbling Robot

A Fraction of a Dot
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Solve the differential equation *(x + y*^{3}) + 6xy^{2}y′ = 0

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

**Incomplete.**

bernoulli :)

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

The case for all x is easier to get from your solution for x > 0: we note that the term with the constant Cx^{-1/2} requires strictly positive x. However, if we want to use all x, we have to set the constant C = 0, so the solution for x > 0 becomes the general solution for all x.