# What is the general solution of the differential equation? : # dy/dx + (2x)/(x^2+1)y=1/(x^2+1) #

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

We have:

This is 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:

Which we can rearrange to get:

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I am going to make a slight change of notation and mark it as equation [1]:

Equation [1] is in the form:

This type of equation is known to have an integrating factor:

Integrate:

Simplify:

Set up both sides for integration:

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This differential equation is completely equivalent to

The homogeneous part of this linear differential equation is

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The general solution of the given differential equation ( \frac{{dy}}{{dx}} + \frac{{2x}}{{x^2+1}}y = \frac{{1}}{{x^2+1}} ) is:

[ y = \frac{{e^{-\int \frac{{2x}}{{x^2+1}} , dx}}}{{x^2+1}} \left( \int \frac{{e^{\int \frac{{2x}}{{x^2+1}} , dx}}}{{x^2+1}} , dx + C \right) ]

where ( C ) is the 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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- Describe the solid whose volume is represented by #int_(0)^(3) (2pi x^5)dx#. ?
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