Answer:
y(x)=0
Explanation:
To solve the given differential equation using Green's function, we need to first determine the Green's function associated with the given boundary conditions.
The Green's function, G(x, ξ), satisfies the following equation:
(x^2 d^2G / dx^2) + (2x dG / dx) - 4G = δ(x - ξ)
where δ(x - ξ) is the Dirac delta function. We can solve this equation subject to the boundary conditions:
G(0, ξ) = G(∞, ξ) = 0
To solve this differential equation, we assume a solution of the form:
G(x, ξ) = A(x)B(ξ)
Substituting this form into the differential equation and simplifying, we get:
x^2 d^2A / dx^2 + 2x dA / dx - 4A = 0
This is a homogeneous second-order ordinary differential equation. We can solve it by assuming a power series solution of the form:
A(x) = ∑[n=0 to ∞] (a_n x^n)
Substituting this series into the differential equation and equating coefficients of like powers of x, we get:
a_n [(n + 2)(n + 1) - 4] = 0
Solving this equation for the coefficients, we find:
a_0 = 0
a_1 = 0
a_n = 4 / [(n + 2)(n + 1)] for n ≥ 2
Therefore, the solution for A(x) is:
A(x) = 4 * ∑[n=2 to ∞] (x^n / [(n + 2)(n + 1)])
Now, we can substitute the solution for A(x) into the form of the Green's function:
G(x, ξ) = A(x)B(ξ)
G(x, ξ) = 4 * ∑[n=2 to ∞] (x^n / [(n + 2)(n + 1)]) * B(ξ)
To determine B(ξ), we impose the boundary conditions:
G(0, ξ) = 0 => 4 * ∑[n=2 to ∞] (0 / [(n + 2)(n + 1)]) * B(ξ) = 0
G(∞, ξ) = 0 => 4 * ∑[n=2 to ∞] (ξ^n / [(n + 2)(n + 1)]) * B(ξ) = 0
From these conditions, we can conclude that B(ξ) = 0. Hence, the Green's function is:
G(x, ξ) = 0
Now, to obtain the solution to the differential equation, we can use the Green's function in the following integral form:
y(x) = ∫[0 to ∞] G(x, ξ) f(ξ) dξ
where f(ξ) is the inhomogeneous term in the original differential equation.
Since G(x, ξ) = 0, the integral evaluates to zero as well. Therefore, the solution to the given differential equation is:
y(x) = 0
In conclusion, the solution to the differential equation with the given boundary conditions is y(x) = 0.