(Solved): Problem 18-) (I) Find the complementary or general solution of the homogeneous differential equatio ...

Problem 18-) (I) Find the complementary or general solution of the homogeneous differential equation \[ x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+16 y=0 \text { for } x>0 \] (II) Find the particular solution of the nonhomogeneous differential equation \[ x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+16 y=-6 \cos (\ln x)+3 \sin (\ln x) \text { for } x>0 \] (III) Find the general solution of the differential equation \[ x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+16 y=-6 \cos (\ln x)+3 \sin (\ln x) \text { for } x>0 \] (IV) Moreover, find the solution of initial value problem of nonhomogeneous differential equation \[ x^{2} \frac{d^{2} y}{d x^{2}}+x \frac{d y}{d x}+16 y=-6 \cos (\ln x)+3 \sin (\ln x) \text { for } x>0, y(1)=2, y^{\prime}(1)=-2 \]