side t cos t has the form with n = 2, α = 0, and β = 1. Since α + iβ = i is a simple root of the characteristic equation we should try a particular integral of the form Substituting yp into the differential equation, we have the identity Comparing both sides, we have which has the solution We then have a particular integral Example 2 Consider the following linear nonhomogeneous differential equation: This is like the first example above, except that the nonhomogeneous part () is not linearly independent to the general solution of the homogeneous part (); as a result, we have to multiply our guess by a sufficiently large power of x to make it linearly independent. Here our guess becomes: By substituting this function and its derivative into the differential equation, one can solve for A: So, the general solution to this differential equation is: Example 3 Find the general solution of the equation: is a polynomial of degree 2, so we look for a solution using the same form, Plugging this particular function into the original equation yields, which gives: Solving for constants we get: To solve for the general solution, where is the homogeneous solution , therefore, the general solution is: References Ordinary differential equations