# What is the general solution of the differential equation? : # Acosx + Bsinx -sinxln|cscx+cotx| #

# Acosx + Bsinx -sinxln|cscx+cotx| #

We have:

Complementary Function

The homogeneous equation associated with [A] is

And it's associated Auxiliary equation is:

Thus the solution of the homogeneous equation is:

Particular Solution

With this particular equation [A], the interesting part is find the solution of the particular function. We would typically use practice & experience to "guess" the form of the solution but that approach is likely to fail here. Instead we must use the Wronskian. It does, however, involve a lot more work:

is given by:

Where:

So for our equation [A]:

So the wronskian for this equation is:

So we form the two particular solution function:

And;

And so we form the Particular solution:

Which then leads to the GS of [A}

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The general solution of the given differential equation is:

[ y(x) = -\sin(x) \ln \left| \csc(x) + \cot(x) \right| + C ]

where ( C ) is an arbitrary constant.

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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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