Problem $1$:

Reduce the ODE to the first order. Show every step in detail.

$2x{y}^{\text{'}\text{'}}=3y^{\text{'}}$

Problem $2$:

Reduce the ODE to the first order. Show every step in detail.

$x{y}^{\text{'}\text{'}}+2y^{\text{'}}+xy=0,y_1=\frac{cosx}{x}$

Problem $3$:

Find a general solution. Check your answer by substitution. Please note, ODEs of

this kind have important applications that are going to be discussed in Secs. $2\mathrm{.}4,$

$2\mathrm{.}7,$ and $2\mathrm{.}9.$

$y^{\text{'}\text{'}}+2k^2{y}^{\text{'}}+k^{4}y=0$

Problem $4$:

Apply the operator to the given functions. Show all steps in detail

$(D-2I{)}^2$;$,e^{2x},x{e}^{2x},e^{-2x}$

Problem $5$:

Find the frequency of oscillation of a pendulum of length

$L,$ neglecting air resistance and the weight of the rod, and

assuming $$ to be so small that $sin()$ practically equals

$.$