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Question Number 125691 by fajri last updated on 13/Dec/20

$$findsolution:$$$$\frac{d^{2}y}{{dx}^2}+4\frac{dy}{dx}=3cosec\theta$$

Answered by mathmax by abdo last updated on 14/Dec/20

$$\mathrm{if}\mathrm{cosec}\theta =\frac{1}{\sin \theta }\mathrm{e}\to {\mathrm{y}}^{{}^{″}}+{4\mathrm{y}}^{{}^{\prime }}=\frac{3}{\sin \theta }(\mathrm{the}\mathrm{variable}\mathrm{is}\mathrm{x}\mathrm{not}\theta !)$$$$\mathrm{let}{\mathrm{y}}^{{}^{\prime }}=\mathrm{z}\Rightarrow {\mathrm{z}}^{{}^{\prime }}+\mathrm{z}=\frac{3}{\sin \theta }$$$$\mathrm{h}\to {\mathrm{z}}^{{}^{\prime }}=-\mathrm{z}\Rightarrow \frac{{\mathrm{z}}^{{}^{\prime }}}{\mathrm{z}}=-1\Rightarrow \mathrm{lnz}=-\mathrm{x}+\mathrm{c}\Rightarrow \mathrm{z}=\mathrm{k}{\mathrm{e}}^{-\mathrm{x}}$$$$\mathrm{lagrange}\mathrm{method}\to {\mathrm{z}}^{{}^{\prime }}={\mathrm{k}}^{{}^{\prime }}{\mathrm{e}}^{-\mathrm{x}}-{\mathrm{ke}}^{-\mathrm{x}}$$$$\mathrm{e}\to {\mathrm{k}}^{{}^{\prime }}{\mathrm{e}}^{-\mathrm{x}}-{\mathrm{ke}}^{-\mathrm{x}}+{\mathrm{ke}}^{-\mathrm{x}}=\frac{3}{\sin \theta }\Rightarrow {\mathrm{k}}^{{}^{\prime }}=\frac{3}{\sin \theta }{\mathrm{e}}^{\mathrm{x}}\Rightarrow$$$$\mathrm{k}=\frac{3}{\sin \theta }\int {\mathrm{e}}^{\mathrm{x}}+\mathrm{c}=\frac{{3\mathrm{e}}^{\mathrm{x}}}{\sin \theta }+\mathrm{c}\Rightarrow \mathrm{z}=(\frac{{3\mathrm{e}}^{\mathrm{x}}}{\sin \theta }+\mathrm{c}){\mathrm{e}}^{-\mathrm{x}}$$$$=\frac{3}{\sin \theta }+{\mathrm{ce}}^{-\mathrm{x}}$$$${\mathrm{y}}^{{}^{\prime }}=\mathrm{z}\Rightarrow {\mathrm{y}}^{{}^{\prime }}=\frac{3}{\sin \theta }+{\mathrm{ce}}^{-\mathrm{x}}\Rightarrow \mathrm{y}=\int (\frac{3}{\sin \theta }+{\mathrm{ce}}^{-\mathrm{x}})\mathrm{dx}\Rightarrow$$$$\mathrm{y}\left(\mathrm{x}\right)=\frac{3\mathrm{x}}{\sin \theta }-\mathrm{c}{\mathrm{e}}^{-\mathrm{x}}+\lambda$$

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