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Re-soudre-2-u-x-2-2-u-y-2-10e-2x-y-




Question Number 163591 by Ar Brandon last updated on 08/Jan/22
Re^� soudre      (∂^2 u/∂x^2 )+(∂^2 u/∂y^2 )=10e^(2x+y)
$$\mathrm{R}\acute {\mathrm{e}soudre}\:\:\:\:\:\:\frac{\partial^{\mathrm{2}} {u}}{\partial{x}^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} {u}}{\partial{y}^{\mathrm{2}} }=\mathrm{10}{e}^{\mathrm{2}{x}+{y}} \\ $$
Commented by mkam last updated on 08/Jan/22
U_p  = (1/(D^2  + D^′ ^2 )) e^(ax + by)    : D^2  + D^′ ^2  ≠ 0    e^(2x + y)  ⇒ a = 2 , b = 1    D(a , b ) = D ( 2 , 1 ) = 2^2  + 1^2  = 5 ≠0    ∴U_p  = ((10)/5) e^(2x + y)  = 2 e^(2x + y)
$$\boldsymbol{{U}}_{{p}} \:=\:\frac{\mathrm{1}}{\boldsymbol{{D}}^{\mathrm{2}} \:+\:\boldsymbol{{D}}^{'} \:^{\mathrm{2}} }\:\boldsymbol{{e}}^{\boldsymbol{{ax}}\:+\:\boldsymbol{{by}}} \:\:\::\:\boldsymbol{{D}}^{\mathrm{2}} \:+\:\boldsymbol{{D}}^{'} \:^{\mathrm{2}} \:\neq\:\mathrm{0} \\ $$$$ \\ $$$$\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}\:+\:\boldsymbol{{y}}} \:\Rightarrow\:\boldsymbol{{a}}\:=\:\mathrm{2}\:,\:\boldsymbol{{b}}\:=\:\mathrm{1} \\ $$$$ \\ $$$$\boldsymbol{{D}}\left(\boldsymbol{{a}}\:,\:\boldsymbol{{b}}\:\right)\:=\:\boldsymbol{{D}}\:\left(\:\mathrm{2}\:,\:\mathrm{1}\:\right)\:=\:\mathrm{2}^{\mathrm{2}} \:+\:\mathrm{1}^{\mathrm{2}} \:=\:\mathrm{5}\:\neq\mathrm{0} \\ $$$$ \\ $$$$\therefore\boldsymbol{{U}}_{{p}} \:=\:\frac{\mathrm{10}}{\mathrm{5}}\:\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}\:+\:\boldsymbol{{y}}} \:=\:\mathrm{2}\:\boldsymbol{{e}}^{\mathrm{2}\boldsymbol{{x}}\:+\:\boldsymbol{{y}}} \\ $$$$ \\ $$
Commented by Ar Brandon last updated on 08/Jan/22
Thank you ����
Commented by mkam last updated on 08/Jan/22
you are welcome sir
$$\boldsymbol{{you}}\:\boldsymbol{{are}}\:\boldsymbol{{welcome}}\:\boldsymbol{{sir}} \\ $$

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