Q25E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Prove that L defined in (7) is a linear operator by verifying that properties (9) and (10) hold for any n-times differentiable functions \({\bf{y,}}\,{{\bf{y}}_{\bf{1}}}{\bf{,}}\,...{\bf{,}}\,{{\bf{y}}_{\bf{m}}}\)  on (a, b).

 

Step-by-Step Solution

Answer

Hence, it is proved that L defined in (7) is a linear operator by verifying that properties (9) and (10) hold for any n-times differentiable functions.

Step 1: Use the properties (9) and (10),

To prove,

 

\({\bf{L}}\left[ {\bf{y}} \right]{\bf{ = }}\frac{{{{\bf{d}}^{\bf{n}}}{\bf{y}}}}{{{\bf{d}}{{\bf{x}}^{\bf{n}}}}}{\bf{ + }}{{\bf{p}}_{\bf{1}}}\frac{{{{\bf{d}}^{{\bf{n - 1}}}}{\bf{y}}}}{{{\bf{d}}{{\bf{x}}^{{\bf{n - 1}}}}}}{\bf{ + }}...{\bf{ + }}{{\bf{p}}_{\bf{n}}}{\bf{y = }}\left( {{{\bf{D}}^{\bf{n}}}{\bf{ + }}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ + }}{{\bf{p}}^{\bf{n}}}} \right)\left[ {\bf{y}} \right]\)

 

is a linear operator by verifying properties (9) and (10).

 

The properties (9) and (10) are given as:

 

\(\begin{array}{l}\left( {\bf{9}} \right).\,\,\,\,\,\,\,\,\,\,\,\,{\bf{L}}\left[ {{{\bf{y}}_{\bf{1}}}{\bf{ + }}{{\bf{y}}_{\bf{2}}}{\bf{ + }}...{\bf{ + }}{{\bf{y}}_{\bf{m}}}} \right]{\bf{ = L}}\left[ {{{\bf{y}}_{\bf{1}}}} \right]{\bf{ + L}}\left[ {{{\bf{y}}_{\bf{2}}}} \right]{\bf{ + }}...{\bf{ + L}}\left[ {{{\bf{y}}_{\bf{m}}}} \right]{\bf{,}}\\\left( {{\bf{10}}} \right).\,\,\,\,\,\,\,\,\,{\bf{L}}\left[ {{\bf{cy}}} \right]{\bf{ = cL}}\left[ {\bf{y}} \right]\end{array}\)

Step 2: Assume that, \({\bf{y,}}\,{{\bf{y}}_{\bf{1}}}{\bf{,}}\,...{\bf{,}}\,{{\bf{y}}_{\bf{m}}}\) is n time differential function on the interval (a, b).

(a, b).

 

Firstly, verify that properties (9) to prove that L is a linear operator. 

 

\(\begin{array}{c}{\bf{L}}\left[ {{{\bf{y}}_{\bf{1}}}{\bf{ + }}{{\bf{y}}_{\bf{2}}}{\bf{ + }}...{\bf{ + }}{{\bf{y}}_{\bf{m}}}} \right]{\bf{ = }}\left( {{{\bf{D}}^{\bf{n}}}{\bf{ + }}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ + }}{{\bf{p}}^{\bf{n}}}} \right)\left[ {{{\bf{y}}_{\bf{1}}}{\bf{ + }}{{\bf{y}}_{\bf{2}}}{\bf{ + }}...{\bf{ + }}{{\bf{y}}_{\bf{m}}}} \right]\\{\bf{ = }}\left( {{{\bf{D}}^{\bf{n}}}{\bf{ + }}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ + }}{{\bf{p}}^{\bf{n}}}} \right)\left[ {{{\bf{y}}_{\bf{1}}}} \right]{\bf{ + }}\left( {{{\bf{D}}^{\bf{n}}}{\bf{ + }}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ + }}{{\bf{p}}^{\bf{n}}}} \right)\left[ {{{\bf{y}}_{\bf{2}}}} \right]{\bf{ + }}...\\{\bf{ + }}\left( {{{\bf{D}}^{\bf{n}}}{\bf{ + }}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ + }}{{\bf{p}}^{\bf{n}}}} \right)\left[ {{{\bf{y}}_{\bf{m}}}} \right]\\{\bf{ = L}}\left[ {{{\bf{y}}_{\bf{1}}}} \right]{\bf{ + L}}\left[ {{{\bf{y}}_{\bf{2}}}} \right]{\bf{ + }}...{\bf{ + L}}\left[ {{{\bf{y}}_{\bf{m}}}} \right]\end{array}\)

 

Step3: Suppose that y is n- time differential function on the interval (a, b).

Now, verifying that properties (10) to prove that L is a linear operator. 

 

\(\begin{array}{c}{\bf{L}}\left[ {{\bf{cy}}} \right]{\bf{ = }}\left( {{{\bf{D}}^{\bf{n}}}{\bf{ + }}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ + }}{{\bf{p}}_{\bf{n}}}} \right)\left[ {{\bf{cy}}} \right]\\{\bf{ = }}{{\bf{D}}^{\bf{n}}}\left[ {{\bf{cy}}} \right]{\bf{ + }}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}\left[ {{\bf{cy}}} \right]{\bf{ + }}...{\bf{ + }}{{\bf{p}}_{\bf{n}}}\left[ {{\bf{cy}}} \right]\\{\bf{ = c}}{{\bf{D}}^{\bf{n}}}\left[ {\bf{y}} \right]{\bf{ + c}}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}\left[ {\bf{y}} \right]{\bf{ + }}...{\bf{ + c}}{{\bf{p}}_{\bf{n}}}\left[ {\bf{y}} \right]\\{\bf{ = c}}\left( {{{\bf{D}}^{\bf{n}}}{\bf{ + }}{{\bf{p}}_{\bf{1}}}{{\bf{D}}^{{\bf{n - 1}}}}{\bf{ + }}...{\bf{ + }}{{\bf{p}}_{\bf{n}}}} \right)\left[ {\bf{y}} \right]\\{\bf{L}}\left[ {{\bf{cy}}} \right]{\bf{ = cL}}\left[ {\bf{y}} \right]\end{array}\)

 

Hence, the differential operator L is a linear operator.