# What is the general solution of the differential equation # dy/dx + y = xy^3 #?

##
Hint: try a substitution # z = 1/y^2#

Hint: try a substitution

# y^2 = 2/(2x + 1 + Ae^(2x) )#

We have:

As suggested we perform a substitution:

And from the chain rule, we have:

Substituting into the initial Differential Equation [A], we have:

This substitution has reduced the equation [A] to a Ordinary Differential Equation of the form, which can be solved using an integrating Factor;

Then the integrating factor is given by;

We can now "separate the variables", to get:

To integrate this integral we will require an application of Integration By Parts:

Then plugging into the IBP formula:

gives us

Using this result, we can now integrate [C] to get:

Restoring the substitution we get:

Which we can write as:

Which is the general solution, or:

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The general solution of the given differential equation ( \frac{dy}{dx} + y = xy^3 ) is:

[ y(x) = \frac{1}{\sqrt{c - x^2 + c x^4}} ]

where ( c ) is a constant of integration.

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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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