A springmass problem has the equation y k2y Coswt Find yH

A spring-mass problem has the equation y\" + k^2y = Coswt. Find y_H Find y_p and also the undetermined constants in y_p. Write the full \"general solution\" For what value of w will the general solution discontinuous.

Solution

An ordinary differential equation is a relation involving one or several derivatives of a function y(x) with respect to x. The relation may also be composed of constants, given functions of x, or y itself. The equation y 0 (x) = e x , (1) where y 0 = dy/dx, is of a first order ordinary differential equation, the equation y 00(x) + 2y(x) = 0, (2) where y 00 = d 2 y/dx2 is of a second order ordinary differential equation, and the equation 2x 2 y 000(x) y 0 (x) + 3e x y 00(x) = (x 2 + 1)y 2 (x), (3) where y 000 = d 3 y/dx3 is a third order ordinary differential equation. The order of an ordinary differential equation the highest derivative of y in the equation. Definition [1]. The explicit solution of a first-order differential equation is a function y = g(x), a < x < b, (4) defined and differentiable on (a, b), with the property that the equation becomes an identity when y and y 0 are replaced by g and g 0 , respectively. The solution of a differential equation G(x, y) = 0 it is called the implicit solution. 1 Example. The explicit solution of the first-order differential equation y 0 (x) = x y(x), (5) is y(x) = c ex 2/2 , (6) where c is an arbitrary constant. The differential equation (5) has many solutions. The function (6), with arbitrary c, represents the general solution (the totality of all solutions of the equation). If we consider a definite value of c, for example c = 1, then the solution obtained y(x) = e x 2/2 is called a particular solution. 2 First order differential equations 2.1 Separable equations The equation g(y)y 0 = f(x), (7) or g(y)dy = f(x)dx, (8) is called an equation with separable variables, or a separable equation. The variable x appears only on the right hand side and the function y appears only on the left hand side in Eq. (8). Integrating both sides we obtain Z g(y)dy = Z f(x)dx + c. (9) If f and g are continuous functions the general solution of Eq. (7) is obtained evaluating Eq. (9).