Question

Find a homogeneous linear differential equation with constant coefficient that is satisfied by y=4xe^xsin2x

Answer 1

$\displaystyle y = 4xe^x\sin(2x)\\ \textsf{If}\,\, y = e^{\alpha x}(A\cos(\beta x) + B\sin(\beta x)),\\ \textsf{the auxiliary equation is}\\ m = \alpha \pm j\beta\\ \therefore m = 1 \pm j2\\ \textsf{The quadratic equation whose roots are}\,\, 1 \pm j2 \,\, \textsf{are}\\ m^2 - (1 + j2 + 1 - j2)m + (1 + j2)(1 - j2) = 0\\ m^2 - 2m + 5 = 0\\ \textsf{For}\,\, e^x\sin(2x) \,\, \textsf{to be multiplied by}\,\, x, \\ \textsf{the roots of}\,\,m\,\,\textsf{must be repeated.}\\ \therefore m = 1 \pm j2 \,\,\textsf{twice}\\ (m^2 - 2m + 5)^2 = 0\\ m^4 - 4m^3 + 14m^2 - 20m + 25 = 0\\ \therefore \textsf{The homogenous differential equation is}\\ y^{(iv)} - 4y^{(iii)} + 14y'' - 20y' + 25y = 0$