Question

We consider the solution of the boundary value problem

(xy′)′+yx=1x,x∈[1,], y(1)=0=y().
by eign function expansion

Answer 1

To solve the boundary value problem using eigenfunction expansion, we assume the solution can be represented as a sum of eigenfunctions and determine the coefficients by applying the boundary conditions. The final solution is obtained by multiplying the eigenfunctions by their respective coefficients.

Step-by-step explanation:

To solve the given boundary value problem using eigenvector expansion, we start by assuming the solution can be represented as a sum of eigenfunctions. Let's assume that the solution can be written in the form y(x) = ∑ AnSn(x), where Sn(x) are the eigenfunctions and An are the coefficients to be determined.

Next, we substitute the assumed solution into the given differential equation and apply the boundary conditions y(1) = 0 and y'(0) = 0 to obtain a set of equations for the coefficients An.

We can then solve this set of equations to determine the values of An. Once we have the values of An, we can express the final solution as a sum of the eigenfunctions multiplied by their respective coefficients.