The Bessel equation provided is second order, linear, has variable coefficients, and is nonhomogeneous. To solve it, one could use variation of parameters or the power series method. The general solution for the homogeneous case is a combination of the given functions y¹(x) and y²(x).
Considering the Bessel equation of one-half order given by the equation x²y′′ + xy′ + (x² − 41) y = x⁵/², for x > 0:
- The differential equation is second order since the highest derivative is y′′ (the second derivative of y with respect to x).
- It is linear in y, y′, and y′′.
- The coefficients are functions of x, making them non-constant.
- This equation is nonhomogeneous due to the term x⁵/².
To solve this differential equation, we could use the method of variation of parameters or the power series method. Moreover, given the specific solutions provided, y¹ (x)=x⁻¹/² cos(x) and y² (x)=x⁻¹/² sin(x), we can state that the homogeneous solution to the original equation would be a linear combination of these two solutions, typically shown as c₁y¹(x) + c₂y²(x), where c₁ and c₂ are constants determined by initial conditions or boundary values.
The general solution to the homogeneous Bessel equation of one-half order will involve a linear combination of the provided functions, resulting in y(x) = c₁(x⁻¹/² cos(x)) + c₂(x⁻¹/² sin(x)).