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Question Number 131887 by Algoritm last updated on 09/Feb/21

Answered by SEKRET last updated on 09/Feb/21

$$\mathbf{Leybnist}\mathbf{formula}$$$$\mathbf{u}={\mathbf{e}}^{-2\mathbf{x}}{\mathbf{u}}^{\mathbf{n}}={(-2)}^{\mathbf{n}}\cdot {\mathbf{e}}^{-2\mathbf{x}}$$$$\mathbf{v}=\frac{1}{\sqrt{1-\mathbf{x}}}{\mathbf{v}}^{\mathbf{n}}=\frac{(2\mathbf{n}-1)!!}{2^{\mathbf{n}}{(1-\mathbf{x})}^{\frac{2\mathbf{n}+1}{2}}}$$$$\mathbf{y}=\mathbf{uv}$$$${\mathbf{y}}^{\mathbf{n}}={\mathbf{C}}_{\mathbf{n}}^0{\mathbf{u}}^{\mathbf{n}}\cdot \mathbf{v}+{\mathbf{C}}_{\mathbf{n}}^1{\mathbf{u}}^{\mathbf{n}-1}\cdot {\mathbf{v}}^1+{\mathbf{C}}_{\mathbf{n}}^2{\mathbf{u}}^{\mathbf{n}-2}\cdot {\mathbf{v}}^2+....+{\mathbf{C}}_{\mathbf{n}}^{\mathbf{n}}{\mathbf{u}}^1\cdot {\mathbf{v}}^{\mathbf{n}}$$

Answered by mathmax by abdo last updated on 10/Feb/21

$$\mathrm{y}\left(\mathrm{x}\right)={\mathrm{e}}^{-2\mathrm{x}}{(1-\mathrm{x})}^{-\frac{1}{2}}\Rightarrow {\mathrm{y}}^{\left(\mathrm{n}\right)}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}^{\mathrm{k}}{\left\{{(1-\mathrm{x})}^{-\frac{1}{2}}\right\}}^{\left(\mathrm{k}\right)}{\left({\mathrm{e}}^{-2\mathrm{x}}\right)}^{(\mathrm{n}-\mathrm{k})}$$$${\mathrm{we}\mathrm{have}{\left({\mathrm{e}}^{-2\mathrm{x}}\right)}^{)\mathrm{n}-\mathrm{k})}={(-2)}^{\mathrm{n}-\mathrm{k}}{\mathrm{e}}^{-2\mathrm{x}}\mathrm{let}\mathrm{find}{(1-\mathrm{x})}^{\mathrm{p}}\}}^{\left(\mathrm{k}\right)}$$$${\left\{{(1-\mathrm{x})}^{\mathrm{p}}\right\}}^{\left(1\right)}=\mathrm{p}(-1){(1-\mathrm{x})}^{\mathrm{p}-1}$$$${\left\{{(1-\mathrm{x})}^{\mathrm{p}}\right\}}^{\left(2\right)}=\mathrm{p}(\mathrm{p}-1){(-1)}^2{(1-\mathrm{x})}^{\mathrm{p}-2}\Rightarrow$$$${\left\{{(1-\mathrm{x})}^{\mathrm{p}}\right\}}^{\left(\mathrm{k}\right)}=\mathrm{p}(\mathrm{p}-1)....(\mathrm{p}-\mathrm{k}+1){(-1)}^{\mathrm{k}}{(1-\mathrm{x})}^{\mathrm{p}-\mathrm{k}}\Rightarrow$$$${\left\{{(1-\mathrm{x})}^{-\frac{1}{2}}\right\}}^{\left(\mathrm{k}\right)}={(-1)}^{\mathrm{k}}(-\frac{1}{2})(-\frac{3}{2})....(-\frac{1}{2}-\mathrm{k}+1){(1-\mathrm{x})}^{-\frac{1}{2}-\mathrm{k}}\Rightarrow$$$${\mathrm{y}}^{\left(\mathrm{n}\right)}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}^{\mathrm{k}}{(-1)}^{\mathrm{k}}(-\frac{1}{2})(-\frac{3}{2})....(-\frac{1}{2}-\mathrm{k}+1){(1-\mathrm{x})}^{-\frac{1}{2}-\mathrm{k}}{(-2)}^{\mathrm{n}-\mathrm{k}}{\mathrm{e}}^{-2\mathrm{x}}$$$$$$

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