# What is the general solution of the differential equation ? # 2xdy/dx = 10x^3y^5+y #

# y = (sqrt(x))/root(4)((C - 4x^5)) #

We have:

We can rewrite as:

This is a Bernoulli equitation which has a standard method to solve. Let:

By the chain rule we have;

Substituting into the last DE we get;

So the substitution has reduced the DE into a first order linear differential equation of the form:

We solve this using an Integrating Factor

Which is now a trivial separable DE, so we can "separate the variables" to get:

And integrating gives us:

Restoring the substitution we get:

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The general solution of the given differential equation is y(x) = [(x^5 - 1) / (5x^4)]^(1/4)

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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