# Solve the differential equation: $$y"" + 2y' + y = \cos x$$

## Question:

Solve the differential equation:

$$y"" + 2y' + y = \cos x$$

## Undetermined Coefficients:

To solve the differential equation $$ay""+by' +cy= f(x)$$ we need to find a the solution to the homogeneous equation {eq}ay""+by' +cy=0{/eq}. This solution will be of the form

$$y_c = C_1 y_1 + C_2 y_2$$

The general solution of this differential equation is an equation of the form

$$y(x) = y_c(x) + y_p(x)$$ where {eq}y_p(x){/eq} is a particular solution. To find a particular solution, we might use the method of undetermined coefficients. We ascertain the form of {eq}y_p(x){/eq} from the form of {eq}g(x){/eq}.

## Answer and Explanation: 1

Become a Study.com member to unlock this answer!

The characteristic equation of our differential equation is

$$r^2+2r+1 =0 = (r+1)^2$$

Second order linear differential equations with repeated roots...

See full answer below.