Читать книгу An Introduction to the Finite Element Method for Differential Equations - Mohammad Asadzadeh - Страница 21

1 Problem 1.1 Consider the problemClearly, the function is a solution. Is this solution unique? Does the answer depend on ?

2 Problem 1.2 Consider the problemIs the solution unique? ( is a given function).Under what condition on a solution exists?

3 Problem 1.3 Suppose are solutions of the linear differential equation , where . Under what condition on the constant coefficients is the linear combination also a solution of this equation?

4 Problem 1.4 Consider the nonlinear ODE .Show that and both are solutions, but is not a solution.For which value of is a solution? What about ?

5 Problem 1.5 Show that each of the following equations has a solution of the form for a proper choice of constants . Find the constants for each example.

6 Problem 1.6Consider the equation . Write the equation in the coordinates .Find the general solution of the equation.Consider the equation . Write it in the coordinates and .

7 Problem 1.7Show that for satisfiesFind a linear combination of s that satisfies .Solve the Dirichlet problem