List the roots of the auxiliary equation for a homogeneous linear differential equation with real, constant coefficients that has the given function as a particular solution. 10) y = x²sinx + Xcosx
Solution to #3 Obtain in factored form a linear differential equation with real, constant coefficients that is satisfied by the given function. 3) y = 2excos3x • The presence of excos3x means that m = 1 ± 3i is a root. So, [D − (1 + 3i)] and [D- (1-3i)] are factors of f(D).
Solution to #3 Obtain in factored form a linear differential equation with real, constant coefficients that is satisfied by the given function. 3) y = 2e*cos3x • Thus, we have [m - (1 + 3i)] [m - (1 − 3i)] = 0 [D- (1 + 3i)] [D - (1-3i)] = 0 [(D 1)² +9] y = 0 (D² - 2D + 10) y = 0