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Question Number 86242 by niroj last updated on 27/Mar/20

$$\mathbf{Find}\mathbf{y}=\mathbf{C}\mathrm{F}+\mathbf{P}\mathrm{I}\mathbf{in}\mathbf{following}\mathbf{differential}\mathbf{equation}:$$$$\frac{{\mathbf{d}}^2\mathbf{y}}{{\mathbf{dx}}^2}+3\frac{\mathbf{dy}}{\mathbf{dx}}+2\mathbf{y}={\mathbf{e}}^{2\mathbf{x}}\mathbf{sinx}.$$$$$$$$$$

Answered by TANMAY PANACEA. last updated on 27/Mar/20

$$y=e^{mx}$$$$m^2{e}^{mx}+3{me}^{mx}+2e^{mx}=0$$$$e^{mx}(m+1)(m+2)=0$$$$e^{mx}\ne 0som=-1,-2$$$$C.F$$$${Ae}^{-x}+{Be}^{-2x}$$$$P.I=\frac{e^{2x}sinx}{D^2+3D+2}$$$$=e^{2x}.\frac{sinx}{{(D+2)}^2+3(D+2)+2}$$$$=e^{2x}.\frac{sinx}{D^2+7D+12}$$$$=e^{2x}.\frac{D^2+12-7D}{{(D^2+12)}^2-49D^2}.sinx$$$$=e^{2x}.\frac{-sinx+12sinx-7cosx}{{(-1^2+12)}^2-49(-1^2)}$$$$=e^{2x}.\frac{11sinx-7cosx}{121+49}=e^{2x}.\frac{11sinx-7cosx}{170}$$$$y={Ae}^{-x}+{Be}^{-2x}+e^{2x}.\frac{11sinx-7cosx}{170}$$

Commented by niroj last updated on 27/Mar/20

$$\mathrm{great}\mathrm{job}\mathrm{sir}.$$

Commented by TANMAY PANACEA. last updated on 27/Mar/20

$$thankyousir$$

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