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

# y(x) = Ae^x+Be^(-x) -1/2xe^x - 1/2 - 1/2e^xln(1+e^x) -1/2 e^-x ln(1+e^x)#

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 (or variation of parameters). 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:

We can evaluate this integral using a substitution:

No we decompose the integrand into partial fractions:

Thus:

And;

We can similarly evaluate this integral using a substitution:

And so we form the Particular solution:

Which then leads to the GS of [A}

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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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