Particular solutions and complementary functions are obtained for the functional equation <mml:math alttext="$\alpha x'\left( t \right) + \beta x'\left( {t - r} \right) + \gamma x\left( t \right) + \delta x\left( {t - r} \right) = f\left( t \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>&#945;</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>&#946;</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mo>&#8242;</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#8722;</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>&#947;</mml:mi><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>&#948;</mml:mi><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>&#8722;</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> in the forms of a convolution type integral and of infinite series.

Solutions of the neutral differential-difference equation <mml:math alttext="$\alpha x'\left( t \right) + \beta x'\left( {t - r} \right) + \gamma x\left( t \right) + \delta x\left( {t - r} \right) = f\left( t \right)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>α</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>β</mml:mi><mml:msup><mml:mi>x</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>γ</mml:mi><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>δ</mml:mi><mml:mi>x</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>r</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>

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International Journal of Mathematics and Mathematical Sciences

Abstract

Copyright

Solutions of the neutral differential-difference equation αx′(t)+βx′(t−r)+γx(t)+δx(t−r)=f(t)

https://www.hindawi.com/journals/ijmms/1992/165083/