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Question Number 112740 by Mr.D.N. last updated on 09/Sep/20

$$\mathrm{Solve}\mathrm{differential}\mathrm{equation}\mathrm{the}\mathrm{method}\mathrm{of}$$$$\mathrm{variation}\mathrm{of}\mathrm{parameters}:$$$$\frac{{\mathbf{d}}^2\mathbf{y}}{{\mathbf{dx}}^2}+4\mathbf{y}=4\mathbf{cosec}2\mathbf{x}$$

Answered by mathmax by abdo last updated on 09/Sep/20

$${\mathrm{y}}^{{}^{″}}+4\mathrm{y}=\frac{4}{\sin \left(2\mathrm{x}\right)}$$$$\mathrm{h}\to {\mathrm{r}}^2+4=0\Rightarrow \mathrm{r}=\stackrel{-}{+}2\mathrm{i}\Rightarrow {\mathrm{y}}_{\mathrm{h}}={\mathrm{ae}}^{2\mathrm{ix}}+{\mathrm{be}}^{-2\mathrm{ix}}=\alpha \cos \left(2\mathrm{x}\right)+\beta \sin \left(2\mathrm{x}\right)$$$$=\alpha {\mathrm{u}}_1+\beta {\mathrm{u}}_2$$$$\mathrm{W}({\mathrm{u}}_1,{\mathrm{u}}_2)=\left|\begin{array}{c}\cos \left(2\mathrm{x}\right)\sin \left(2\mathrm{x}\right)\\ -2\sin 2\mathrm{x}2\cos \left(2\mathrm{x}\right)\end{array}\right|=2({\cos }^{2}2\mathrm{x}+{\sin }^2\left(2\mathrm{x}\right))=2\ne 0$$$${\mathrm{W}}_1=\left|\begin{array}{c}0\sin \left(2\mathrm{x}\right)\\ \frac{4}{\sin \left(2\mathrm{x}\right)}2\cos \left(2\mathrm{x}\right)\end{array}\right|=-4$$$${\mathrm{W}}_2=\left|\begin{array}{c}\cos \left(2\mathrm{x}\right)0\\ -2\sin \left(2\mathrm{x}\right)\frac{4}{\sin \left(2\mathrm{x}\right)}\end{array}\right|=4\mathrm{cotan}\left(2\mathrm{x}\right)$$$${\mathrm{V}}_1=\int \frac{{\mathrm{W}}_1}{\mathrm{W}}\mathrm{dx}=\int \frac{-4}{2}\mathrm{dx}=-2\mathrm{x}$$$${\mathrm{V}}_2=\int \frac{{\mathrm{W}}_2}{\mathrm{W}}\mathrm{dx}=\int \frac{4\mathrm{cotan}\left(2\mathrm{x}\right)}{2}\mathrm{dx}=2\int \frac{\cos \left(2\mathrm{x}\right)}{\sin \left(2\mathrm{x}\right)}\mathrm{dx}=\ln (\mid \sin \left(2\mathrm{x}\right)\mid )$$$$\Rightarrow {\mathrm{y}}_{\mathrm{p}}={\mathrm{u}}_1{\mathrm{v}}_1+{\mathrm{u}}_2{\mathrm{v}}_2=\cos \left(2\mathrm{x}\right)(-2\mathrm{x})+\sin \left(2\mathrm{x}\right)\ln \mid \sin \left(2\mathrm{x}\right)\mid$$$$\mathrm{the}\mathrm{general}\mathrm{solution}\mathrm{is}\mathrm{y}={\mathrm{y}}_{\mathrm{h}}+{\mathrm{y}}_{\mathrm{p}}$$$$\mathrm{y}=\alpha \cos \left(2\mathrm{x}\right)+\beta \sin \left(2\mathrm{x}\right)-2\mathrm{x}\cos \left(2\mathrm{x}\right)+\sin \left(2\mathrm{x}\right)\ln \mid \sin \left(2\mathrm{x}\right)\mid$$

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