## Methods of applied mathematics |

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applies arbitrary associated augmented matrix calculation calculus of variations characteristic functions characteristic numbers characteristic value coefficient matrix column components considered constant constraint continuous solution converges coordinates corresponding characteristic vectors deduce defined deflection denote determined differential equation end conditions equivalent Euler equation exists expressed follows form y(x Fredholm equation function y(x given Green's function hence Hermitian homogeneous infinite integrand interval involved iterative kernel K(x Lagrange multipliers left-hand member linear combination linearly independent multiple natural boundary conditions nonsingular nontrivial solution normal notation notice obtained orthogonal matrix positive definite possesses potential energy prescribed procedure quadratic form quantities reduced relation relevant replaced result of Problem right-hand member satisfies the equation scalar product set of equations square matrix stationary function Suppose symmetric matrix takes the form transformation unit vectors vanish variables variational problem verify zero