Solve y′′+2y′+4y = 13cos(4x−2) using method of undeterminedcoefficients

To solve the given second order non-homogeneous differential equation using the method of undetermined coefficients, we first need to find the solution of the corresponding homogeneous equation, then find a particular solution of the non-homogeneous equation, and finally, add these two solutions to get the general solution.

Step 1: Solve the homogeneous equation The homogeneous equation is y'' + 2y' + 4y = 0. This is a second order linear homogeneous differential equation with constant coefficients. The characteristic equation is r^2 + 2r + 4 = 0. Solving this quadratic equation, we get r = -1 ± √3 i. Therefore, the general solution of the homogeneous equation is y_h(x) = e^(-x)(A cos(√3 x) + B sin(√3 x)).

Step 2: Find a particular solution of the non-homogeneous equation The non-homogeneous term is 13cos(4x - 2). We guess a solution in the form y_p(x) = C cos(4x - 2) + D sin(4x - 2). Differentiate y_p(x) twice and substitute into the non-homogeneous equation to find the coefficients C and D.

Step 3: Add the homogeneous and particular solutions The general solution of the non-homogeneous equation is y(x) = y_h(x) + y_p(x).