An alternative method is shown for solving the differential equation <mml:math alttext="$y^{\left( k \right)}  - f\left( x \right)y = 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>&#8722;</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> by means of series. Also included is a result for a sequence of functions <mml:math alttext="$\left\{ {S_n \left( x \right)} \right\}_{n = 1}^\infty  $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>&#8734;</mml:mi></mml:msubsup></mml:mrow></mml:math> which gives conditions under which <mml:math alttext="$\mathop {\lim }\limits_n \left( {\frac{{d^k }} {{dx^k }}S_n \left( x \right)} \right) = \frac{{d^k }} {{dx^k }}\left( {\mathop {\lim }\limits_n S_n \left( x \right)} \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:munder><mml:mrow><mml:mo>lim</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:munder><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:msub><mml:mi>S</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow><mml:mrow><mml:mi>d</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mi>k</mml:mi></mml:msup></mml:mrow></mml:mfrac><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:munder><mml:mrow><mml:mo>lim</mml:mo></mml:mrow><mml:mi>n</mml:mi></mml:munder><mml:msub><mml:mi>S</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>.

A method of solving <mml:math alttext="$y^{\left( k \right)}  - f\left( x \right)y = 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>y</mml:mi><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>k</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:msup><mml:mo>−</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>y</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>

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International Journal of Mathematics and Mathematical Sciences

A method of solving y(k)−f(x)y=0

Abstract

Copyright