- #1
Jhenrique
- 685
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Given the following diff equation: ##(1 - D)y(x) = f(x)##, being D = d/dx, the "implicit" solution is: ##y(x) = \frac{1}{1-D}f(x)##, so, for "explicit" the solution is necessary to expand the fraction 1/(1-D) by identity: ##\frac{1}{1-x}=\sum_{0}^{\infty}x^n \Delta n##, but this infinity series is true only for |x|<1, for |x|>1 is necessary utilize the following series: ##\frac{1}{1-x} = -\sum_{-\infty}^{-1} x^n \Delta n##. Happens that D isn't a number for I say that |D| is less or greater than 1. So, how can I interprate this form of solution correctly?
PS, this ideia from Operational Calculus (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/H.-J.%20Glaeske,%20A.%20P.%20Prudnikov,%20K.A.%20Skornik%20-%20Operational%20Calculus%20and%20Related%20Topics%20%28Analytical%20Methods%20and%20Special%20Functions%29%20%28Chapman%20,2006%29.pdf )
PS, this ideia from Operational Calculus (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/H.-J.%20Glaeske,%20A.%20P.%20Prudnikov,%20K.A.%20Skornik%20-%20Operational%20Calculus%20and%20Related%20Topics%20%28Analytical%20Methods%20and%20Special%20Functions%29%20%28Chapman%20,2006%29.pdf )
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