Question and Answers Forum

All Questions      Topic List

Differential Equation Questions

Previous in All Question      Next in All Question      

Previous in Differential Equation      Next in Differential Equation      

Question Number 43616 by Tawa1 last updated on 12/Sep/18

$$\mathrm{Solve}:{\mathrm{x}}^2{\mathrm{y}}^{″}+{\mathrm{xy}}^{\prime }-\mathrm{y}=\mathrm{x},{\mathrm{y}}_1=\mathrm{x}$$

Answered by tanmay.chaudhury50@gmail.com last updated on 13/Sep/18

$$x=e^t\frac{dx}{dt}=e^t$$$$\frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}=\frac{dy}{dt}\times \frac{1}{e^t}$$$$\frac{dy}{dt}=e^t.\frac{dy}{dx}=x\frac{dy}{dx}$$$$\frac{d^{2}y}{{dx}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dt}\left(\frac{1}{e^t}\frac{dy}{dt}\right)\times \frac{dt}{dx}$$$$=(\frac{1}{e^t}\times \frac{d^{2}y}{{dt}^2}-e^{-t}\times \frac{dy}{dt})\times \frac{dt}{dx}$$$$=\frac{1}{e^t}(\frac{d^{2}y}{{dt}^2}-\frac{dy}{dt})\times \frac{1}{e^t}$$$${\left(e^t\right)}^2\times \frac{d^{2}y}{{dx}^2}=(\frac{d^{2}y}{{dt}^2}-\frac{dy}{dt})$$$$x^2\times \frac{d^{2}y}{{dx}^2}=\frac{d^{2}y}{{dt}^2}-\frac{dy}{dt}$$$$nowsubstitutingthevalueobtained$$$$\frac{d^{2}y}{{dt}^2}-\frac{dy}{dt}+\frac{dy}{dt}-y=e^t$$$$\frac{d^{2}y}{{dt}^2}-y=e^t$$$$lety=e^{mt}isasolution$$$$m^2{e}^{mt}-e^{mt}=0$$$$e^{mt}(m^2-1)=0$$$$e^{my}cannotbezero$$$$m=\pm 1$$$$c.F={Ae}^t+{Be}^{-t}$$$$P.I=\frac{e^t}{D^2-1}$$$$=\frac{e^t}{(D+1)(D-1)}$$$$=\frac{1}{(1+1)}\times \frac{e^t}{}\times \frac{1}{D+1-1}.1$$$$=\frac{e^t}{2}\times t$$$$completesolution$$$$={Ae}^t+B^{}{e}^{-t}+\frac{{te}^t}{2}$$$$=Ax+{Bx}^{-1}+\frac{x\times lnx}{2}$$$$$$$$$$

Commented by Tawa1 last updated on 13/Sep/18

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com