# What is a second solution to the Differential Equation # x^2y'' -3xy'+5y=0 #?

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We are given that the answer does not contain cosine functions.

We are given that the answer does not contain cosine functions.

See below.

Assuming that the differential equation reads

The differential equation

Proposing

then

now according to

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Second solution is:

#y=Bx^2sin(lnx)#

If we assume the a corrected equation:

This is a Euler-Cauchy Equation which is typically solved via a change of variable. Consider the substitution:

Then we have,

Substituting into the initial DE [A] we get:

This is now a second order linear homogeneous Differentiation Equation. The standard approach is to look at the Auxiliary Equation, which is the quadratic equation with the coefficients of the derivatives, i.e.

We can solve this quadratic equation, and we get two complex conjugate roots:

Thus the Homogeneous equation [B]:

has the solution:

Now we initially used a change of variable:

So restoring this change of variable we get:

Which is the General Solution.

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The second solution to the differential equation (x^2y'' - 3xy' + 5y = 0) can be found by using the method of Frobenius. This method is used for finding solutions to second-order linear differential equations with variable coefficients.

First, we assume a solution of the form (y = \sum_{n=0}^\infty a_nx^{n+r}), where (r) is a constant to be determined and (a_n) are coefficients.

Then, we substitute this form into the given differential equation and solve for (r) by collecting like terms and setting the coefficient of each power of (x) to zero.

Once we find (r), we can determine the values of (a_n) using a recurrence relation, and this will give us the second linearly independent solution.

Without solving the differential equation, it's not possible to provide the exact form of the second solution.

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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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