(Solved): (a) Consider the following differential equation \( x^{2} \frac{d^{2} y}{d x^{2}}-4 x \frac{d y}{d ...

(a) Consider the following differential equation \( x^{2} \frac{d^{2} y}{d x^{2}}-4 x \frac{d y}{d x}+6 y=7 x^{4} \sin x \). (i) Determine the type of the equation given. Justify your answer. (ii) Given the complementary solution is of the form \( y_{c}=c_{1} x^{3}+c_{2} x^{2} \) where \( c_{1} \) and \( c_{2} \) are arbitrary constants, use the method of variation of parameters to find a particular solution \( y_{p} \) of the above equation. (b) Consider the following differential equation \( x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=x^{4} e^{x} \). (i) Explain why the particular solution of the equation is obtained using variation of parameters. (ii) Find the general solution to the equation.