(Solved): Problem 2. A second-order nonhomogeneous linear differential equation with constant coefficients ( ...

Problem 2. A second-order nonhomogeneous linear differential equation with constant coefficients (NHLDECC) with the nonhomogeneous term \( g(x)=2 \cosh 2 x \), its associated HLDECC, the polynomial equation (APE) associated with the HLDECC, and the general solution of the NHLDECC The second problem. a) Find the general solution of the homogeneous, linear differential equations with constant coefficients (HLDECC) \[ \frac{d^{2} y}{d x^{2}}+6 \frac{d y}{d x}+9 y=0 \] b) Apply the method of undetermined coefficients to find a particular solution of the nonhomogeneous, linear differential equations with constant coefficients (NHLDECC) \[ \frac{d^{2} y}{d x^{2}}+6 \frac{d y}{d x}+9 y=2 \cosh 2 x . \] Recall that \( \cosh 2 x=\frac{e^{2 x}+e^{-2 x}}{2} \) and \( \sinh 2 x=\frac{e^{2 x}-e^{-2 x}}{2} \) so that \[ 2 \cosh 2 x=e^{2 x}+e^{-2 x} \] c) Find the general solution of the homogeneous, linear differential equations with constant coefficients (NHLDECC) \[ \frac{d^{2} y}{d x^{2}}+6 \frac{d y}{d x}+9 y=2 \cosh 2 x \]