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 86142 by jagoll last updated on 27/Mar/20

$${\mathrm{y}}^{\prime }.\sin \mathrm{t}\cos \mathrm{t}=\mathrm{y}+\sin {}^3\mathrm{t}$$$$\mathrm{y}\left(\frac{\pi }{4}\right)=0$$

Answered by Kunal12588 last updated on 27/Mar/20

$$\frac{dy}{dt}=\frac{y}{\sin t\cos t}+\sin t\tan t$$$$\Rightarrow \frac{dy}{dt}+(-\frac{1}{\sin t\cos t})y=\sin t\tan t$$$$P=-\frac{1}{\sin t\cos t};Q=\sin t\tan t$$$$I.F=e^{\int Pdt}=e^{-\int \frac{dt}{\sin t\cos t}}=e^{-\int \frac{{\sin }^{2}t+{\cos }^{2}t}{\sin t\cos t}dt}$$$$=e^{-\int \tan tdt-\int \cot tdt}=e^{-\ln \mid \sec t\mid -\ln \mid \sin t\mid }$$$$=e^{-\ln \mid \tan t\mid }=e^{\ln \mid \cot t\mid }=\cot t$$$$y(I.F)=\int Q(I.F)dt$$$$\Rightarrow y\cot t=\int \sin tdt$$$$\Rightarrow y\cot t=-\cos t+a$$$$\Rightarrow y=-\sin t+a\tan t\to generalsolution$$$$Ifquestionmeanswhent=\frac{\pi }{4},y=0.Then,$$$$0=-\frac{1}{\sqrt{2}}+a$$$$\Rightarrow a=\frac{1}{\sqrt{2}}$$$$\therefore y=-\sin t+\frac{1}{\sqrt{2}}\tan t\to particularsolution$$

Commented by jagoll last updated on 27/Mar/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}$$

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

$$\frac{dy}{dt}+\frac{-y}{sintcost}=\frac{{sin}^{2}t}{cost}$$$$e^{\int \frac{dt}{-sintcost}}$$$$e^{\int -2cosec2tdt}$$$$\int cosec2t=\frac{lntant}{2}$$$$e^{\int -2cosec2t}dt=e^{-2\times \frac{lntant}{2}}=\frac{1}{tant}=cott$$$$cott\times \frac{dy}{dt}+\frac{-y\times cott}{sint\times cost}=\frac{{sin}^{2}t}{cost}\times cott$$$$cott\times \frac{dy}{dt}+y\times (-{cosec}^{2}t)=sint$$$$\frac{d}{dt}(y.cott)=sint$$$$ycott=-cost+c$$$$0\times 1=-\frac{1}{\sqrt{2}}+c$$$$ycott=-cost+\frac{1}{\sqrt{2}}$$

Commented by jagoll last updated on 27/Mar/20

$$\mathrm{what}\mathrm{is}\mathrm{cott}\mathrm{sir}?$$

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

$$cot\theta \to cot\left(t\right)$$

Commented by jagoll last updated on 27/Mar/20

$$\mathrm{ol}\mathrm{it}\cot \left(\mathrm{t}\right)..\mathrm{thank}\mathrm{you}\mathrm{sir}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com