Question and Answers Forum

All Questions      Topic List

Others Questions

Previous in All Question      Next in All Question      

Previous in Others      Next in Others      

Question Number 88235 by Rio Michael last updated on 09/Apr/20

$$\mathrm{find}\mathrm{a}\mathrm{maclaurine}\mathrm{series}\mathrm{solution}\mathrm{to}\mathrm{the}\mathrm{differential}\mathrm{equation}$$$$\mathrm{up}\mathrm{to}\mathrm{the}\mathrm{term}\mathrm{in}{x}^4.$$$$\frac{dy}{dx}-x=xy\mathrm{if}y=1\mathrm{when}x=0.$$

Commented by niroj last updated on 11/Apr/20

$$\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{x}=\mathrm{xy}\mathrm{if}\mathrm{y}=1\mathrm{when}\mathrm{x}=0.$$$$\mathrm{by}\mathrm{the}\mathrm{help}\mathrm{of}\mathrm{linear}\mathrm{diff}..{\mathrm{equ}}^{\mathrm{n}}\mathrm{will}\mathrm{be}..$$$$\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{x}-\mathrm{xy}=0$$$$\frac{\mathrm{dy}}{\mathrm{dx}}-\mathrm{xy}=\mathrm{x}$$$$\mathrm{P}=-\mathrm{x}\mathrm{\&}\mathrm{Q}=\mathrm{x}$$$$\mathrm{IF}={\mathrm{e}}^{\int \mathrm{Pdx}}={\mathrm{e}}^{-\frac{{\mathrm{x}}^2}{2}}$$$$\mathrm{y}\times \mathrm{IF}=\int \mathrm{IF}\times \mathrm{Qdx}+\mathrm{C}$$$$\mathrm{y}.{\mathrm{e}}^{-\frac{{\mathrm{x}}^2}{2}}=\int {\mathrm{e}}^{-\frac{{\mathrm{x}}^2}{2}}.\mathrm{xdx}+\mathrm{C}$$$$\mathrm{put}{\mathrm{x}}^2=\mathrm{t}$$$$2\mathrm{xdx}=\mathrm{dt}$$$$\mathrm{xdx}=\frac{\mathrm{dt}}{2}$$$$\mathrm{y}.{\mathrm{e}}^{-\frac{{\mathrm{x}}^2}{2}}=\int {\mathrm{e}}^{-\frac{\mathrm{t}}{2}}.\frac{\mathrm{dt}}{2}+\mathrm{C}$$$$\mathrm{y}.{\mathrm{e}}^{-\frac{{\mathrm{x}}^2}{2}}=\frac{1}{2}\int {\mathrm{e}}^{-\frac{\mathrm{t}}{2}}\mathrm{dt}+\mathrm{C}$$$${\mathrm{ye}}^{-\frac{{\mathrm{x}}^2}{2}}=\frac{1}{2}.\frac{{\mathrm{e}}^{-\frac{\mathrm{t}}{2}}}{-\frac{1}{2}}+\mathrm{C}$$$${\mathrm{ye}}^{-\frac{{\mathrm{x}}^2}{2}}=-.{\mathrm{e}}^{-\frac{{\mathrm{x}}^2}{2}}+\mathrm{C}$$$${\mathrm{ye}}^{-\frac{{\mathrm{x}}^2}{2}}+{\mathrm{e}}^{-\frac{{\mathrm{x}}^2}{2}}=\mathrm{C}$$$$\mathrm{if}\mathrm{y}=1\mathrm{and}\mathrm{x}=0$$$$1.{\mathrm{e}}^{-\frac{0}{2}}+{\mathrm{e}}^{\frac{-0}{2}}=\mathrm{C}$$$$\mathrm{C}=1.1+1=2$$$$\therefore {\mathrm{ye}}^{-\frac{{\mathrm{x}}^2}{2}}+{\mathrm{e}}^{-\frac{{\mathrm{x}}^2}{2}}=2$$$$\frac{\mathrm{y}+1}{{\mathrm{e}}^{\frac{{\mathrm{x}}^2}{2}}}=2$$$$\mathrm{y}={2\mathrm{e}}^{\frac{{\mathrm{x}}^2}{2}}-1//.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com