Answer :
The Legendre polynomials Po(x), P1(x), P2(x), P3(x), P4(x), and P5(x) are given. To find the first three positive eigenvalues (values of λ) for which the problem (1-x²)y" - 2xy + 2y = λy, subject to the boundary conditions y(0) = 0 and y(x), y(x) bounded on [-1,1], has nontrivial solutions, we need to determine the roots of the characteristic equation corresponding to this problem. The eigenvalues λ that satisfy this condition are 2, 6, and 12.
To find the first three positive values of n for which the given boundary value problem has nontrivial solutions, we need to match the characteristic equation of the Legendre polynomials with the differential equation provided.
The given differential equation is:
[tex](1 - x^2)y" - 2xy + 2y = 0[/tex]
To solve this equation, we can assume a power series solution [tex]y(x) = \sum(a_nx^n),[/tex]and substitute it into the equation.
After simplification, we obtain the following equation involving the coefficients [tex]a_n:[/tex]
[tex]n(n-1)a_nx^{(n-2)} - 2a_nx^n + 2a_nx^n = 0[/tex]
This simplifies to:
[tex]n(n-1)a_n - 2a_n = 0[/tex]
Solving for a_n, we get:
[tex]a_n = 0[/tex] (for n > 2)
This implies that the only nonzero coefficients occur for n = 0 and n = 1.
Now, let's consider the boundary conditions:
y(0) = 0 => a_0 = 0
y'(x) = 0 at x = ±1
The condition y'(x) = 0 at x = ±1 implies that a_1 must be 0 as well.
Since we are looking for nontrivial solutions (solutions other than y(x) = 0), we need to find values of n for which a_n is nonzero.
Therefore, we are interested in the Legendre polynomials where a_n ≠ 0.
The first Legendre polynomial with nonzero coefficient is [tex]P2(x) = 3x^2 - 1.[/tex] Substituting n = 2 into the characteristic equation gives:
[tex]2(2 - 1)a_2 - 2a_2 = 0[/tex]
[tex]a_2[/tex] ≠ 0
Thus, the first nontrivial solution occurs for n = 2.
For n = 3 and beyond, the Legendre polynomials have coefficients [tex]a_n = 0,[/tex] so they do not provide nontrivial solutions to the given problem.
In conclusion, the first three positive values of n for which the given problem has nontrivial solutions are n = 2.
For similar question on Legendre polynomials.
https://brainly.com/question/33067480
#SPJ11
