# What is the general solution of the differential equation? # x^2y'' +3xy'+17y=0 #

Assuming that the differential equation reads

proposing a solution with the structure

Solving now

but

so we can reduce the solutions to the form

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# y=(Acos(4lnx))/x+(Bsin(4lnx))/x#

We have:

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 general solution of the differential equation (x^2y'' + 3xy' + 17y = 0) can be found by assuming a solution of the form (y = x^r), where (r) is a constant. Substituting this into the equation gives a characteristic equation, which can be solved to find the roots (r_1) and (r_2). The general solution is then given by:

[y = C_1x^{r_1} + C_2x^{r_2}]

where (C_1) and (C_2) are arbitrary constants, and (r_1) and (r_2) are the roots of the characteristic equation.

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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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