Question

If ∑=0[infinity] y=∑ k=0is a solution of the differential equation (2−3)y′′+2=0, then its coefficients are related by the equation:

a) 2aₖ+2−3aₖ+2=0
b) 2aₖ−2−3aₖ+2=0
c) 2aₖ+2−3aₖ+1+2=0
d) 2aₖ−2−3aₖ−1+2=0

Answer 1

The differential equation (2-3)y''+2=0 suggests a relation between the coefficients a_k of the power series solution y. Upon substitution and comparison of coefficients, the correct relationship is found to be 2a_k+2-3a_k+1+2=0.

Step-by-step explanation:

The given differential equation is (2−3)y′′+2=0. We need to find out how the coefficients of the series solution are related. Given that y=∞Σk=0 a_kx^{k}, we can plug this series into the differential equation and align terms of the same power of x.

We know that the second derivative of y with respect to x, y′′, will involve terms like k(k-1)a_kx^{k-2}. When substituting into the equation, we adjust the index to match the powers of x.

By comparing coefficients, we find that for each power of x, the coefficients must satisfy a specific relation to zero out the terms as required by the differential equation. The correct relationship is c) 2aₖ+2−3aₖ+1+2=0.