Use the mass–spring oscillator analogy to decide whether all solutions to each of the following differential equations are bounded as \({\bf{t}} \to {\bf{ + }}\infty \)

\(\begin{array}{l}\left( {\bf{a}} \right)\,\,{\bf{y'' + }}{{\bf{t}}^{\bf{4}}}{\bf{y = 0}}\\\left( {\bf{b}} \right)\,\,{\bf{y'' - }}{{\bf{t}}^{\bf{4}}}{\bf{y = 0}}\\\left( {\bf{c}} \right)\,\,{\bf{y'' + }}{{\bf{y}}^{\bf{7}}}{\bf{ = 0}}\\\left( {\bf{d}} \right)\,\,{\bf{y'' + }}{{\bf{y}}^{\bf{8}}}{\bf{ = 0}}\\\left( {\bf{e}} \right)\,\,{\bf{y'' + }}\left( {{\bf{3 + sint}}} \right){\bf{y = 0}}\\\left( {\bf{f}} \right)\,\,{\bf{y'' + }}{{\bf{t}}^{\bf{2}}}{\bf{y' + y = 0}}\\\left( {\bf{g}} \right)\,\,{\bf{y'' - }}{{\bf{t}}^{\bf{2}}}{\bf{y' - y = 0}}\end{array}\)

Given a differential equation,

\({\bf{y'' + }}{{\bf{t}}^{\bf{4}}}{\bf{y = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( {\bf{1}} \right)\)

One knows that,

\({\bf{my'' + by' + ky = 0}}\)

Compare the equation (1) with the above equation,

\({\bf{k = }}{{\bf{t}}^{\bf{4}}}{\bf{ > 0}}\)

Therefore, the given differential equation is bounded.

Given a differential equation,

\({\bf{y'' - }}{{\bf{t}}^{\bf{4}}}{\bf{y = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( {\bf{2}} \right)\)

One knows that,

\({\bf{my'' + by' + ky = 0}}\)

Compare the equation (2) with the above equation,

\({\bf{k = - }}{{\bf{t}}^{\bf{4}}}{\bf{ < 0}}\)

Therefore, the given differential equation is unbounded.

Given differential equation,

\({\bf{y'' + }}{{\bf{y}}^{\bf{7}}}{\bf{ = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( {\bf{3}} \right)\)

One knows that,

\({\bf{my'' + by' + ky = 0}}\)

Compare the equation (3) with the above equation,

\({\bf{k = }}{{\bf{t}}^{\bf{6}}}{\bf{ > 0}}\)

Therefore, the given differential equation is bounded.

Given a differential equation,

\({\bf{y'' + }}{{\bf{y}}^{\bf{8}}}{\bf{ = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( {\bf{4}} \right)\)

One knows that,

\({\bf{my'' + by' + ky = 0}}\)

Compare the equation (4) with the above equation,

\({\bf{k = }}{{\bf{t}}^{\bf{7}}}{\bf{ > 0}}\)if \({\bf{y < 0}}\)

Therefore, the given differential equation is unbounded.

Given a differential equation,

\({\bf{y'' + }}\left( {{\bf{3 + sint}}} \right){\bf{y = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( {\bf{5}} \right)\)

One knows that,

\({\bf{my'' + by' + ky = 0}}\)

Compare the equation (5) with the above equation,

\({\bf{k = 3 + sint > 0}}\)

Therefore, the given differential equation is bounded.

Given differential equation,

\({\bf{y'' + }}{{\bf{t}}^{\bf{2}}}{\bf{y' + y = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( {\bf{6}} \right)\)

We know that,

\({\bf{my'' + by' + ky = 0}}\)

Compare the equation (6) with above equation,

\({\bf{k = 1 > 0}}\)

Therefore, the given differential equation is bounded.

Given a differential equation,

\({\bf{y'' - }}{{\bf{t}}^{\bf{2}}}{\bf{y' - y = 0}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( {\bf{7}} \right)\)

One knows that,

\({\bf{my'' + by' + ky = 0}}\)

Compare the equation (7) with above equation,

\({\bf{k = - 1 < 0}}\)

Therefore, given differential equation is unbounded.