# ODE4.1

First, solve the homogeneous equation to get . Then, guess a particular solution and add them together to get the general solution .

Set up the characteristic equation and solve.

So,

Therefore, the solution to the homogeneous equation is

Now, check the homogeneous solution to see if any new information can be gained. Start by taking the first and second derivatives of .

Substitution of these derivatives into the homogeneous version of the original differential equation and collection of the exponential and trig functions gives the following relations.

So from the first equation, is arbitrary and , so . From the second equation, no new information is obtained. Therefore, a better homogeneous equation is

Now a particular solution should be guessed. Based on the nonhomogeneous part of the original DE, , a first guess is

, with , , and undetermined constants.

But, the homogeneous solution appears as one of the terms of the guessed particular solution, so that term should be multiplied by . Now, a guess for the particular solution is

Find the first and second derivatives of .

Substitute these derivatives into the original DE to get these relations:

So comparing the coefficients of and gives no information about except that it is arbitrary so far. The other terms give the relations

so that , and

is arbitrary, so that the particular solution is

Finally, the general solution is