Checking Qn(x) is a solution to legendre eq. (n=3)

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In summary, In order to verify that Q3(x) is a solution to the legendre equation with n=3, you can use the fact that the expression for P3(x) is a factor in the last term and satisfies the differential equation. This can save time and effort compared to substituting the expression directly into the equation.
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linda300
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Hey,

I have a question which ends by asking to verify that Q3(x) is a solution to the legendre equation,

I took the first and second derivatives of it and before I continue with this messy verification I wanted to know if there was a simpler way to check.

Q3(x) = (1/4)x(5x^2 - 3)log((1+x)/(1-x)),

Is there some quick way to verify its a solution to the n=3 legendre equation? Rather then substituting it straight into the equation?
 
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  • #2
Your expression for Q3(x) seems to be missing a few terms.

Note that ##P_3(x) = \frac{1}{2}(5x^3-3x)## is a factor in the last term. Instead of multiplying everything out and grinding it out, use the fact that P3(x) satisfies the differential equation.
 

FAQ: Checking Qn(x) is a solution to legendre eq. (n=3)

How do I check if Qn(x) is a solution to the Legendre equation for n=3?

To check if Qn(x) is a solution to the Legendre equation for n=3, you can plug the function into the equation and see if it satisfies the equation. The Legendre equation for n=3 is (1-x^2)y'' - 2xy' + 3y = 0.

What is the Legendre equation and why is it important to check the solutions?

The Legendre equation is a differential equation that arises in many areas of mathematics and physics. Its solutions, known as Legendre polynomials, have many important applications in areas such as quantum mechanics, electromagnetics, and approximation theory. It is important to check the solutions to ensure that they satisfy the equation and are valid solutions.

Can I use any method to check if Qn(x) is a solution to the Legendre equation?

Yes, there are multiple methods you can use to check if Qn(x) is a solution to the Legendre equation. One method is to directly plug the function into the equation and see if it satisfies it. Another method is to use properties of Legendre polynomials, such as orthogonality, to verify the solution.

What are some properties of Legendre polynomials that can be used to check solutions?

Some properties of Legendre polynomials that can be used to check solutions include orthogonality, which states that the integral of the product of two Legendre polynomials is zero, and recurrence relations, which express Legendre polynomials of different orders in terms of each other.

Are there any other equations that Qn(x) must satisfy to be a valid solution to the Legendre equation?

Yes, in addition to satisfying the Legendre equation (1-x^2)y'' - 2xy' + n(n+1)y = 0, Qn(x) must also satisfy certain boundary conditions. These conditions vary depending on the specific problem being solved, but commonly include the requirement that the solution is finite at all points and that it satisfies certain boundary conditions at the endpoints of the interval of interest.

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