Sturm–Liouville Form. A second-order equation is said to be in Sturm–Liouville form if it is expressed as $\left[p\left(t\right)y\text{'}\left(t\right)\right]\mathbf{\text{'}}\mathbf{+}\mathbf{q}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$. Show that the substitutions${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{y}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{py}\mathbf{\text{'}}$ result in the normal form $\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{1}}\mathbf{=}\frac{{\mathbf{x}}_{\mathbf{2}}}{\mathbf{p}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{-}{\mathbf{qx}}_{\mathbf{1}}$. If$\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{a}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{b}$ are the initial values for the Sturm–Liouville problem, what are ${\mathbf{x}}_{\mathbf{1}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{and}{\mathbf{x}}_{\mathbf{2}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}$?

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