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 105569 by bemath last updated on 30/Jul/20

$$x^2\frac{dy}{dx}-3xy-2y^2=0$$

Answered by john santu last updated on 30/Jul/20

$$\frac{dy}{dx}=\frac{3xy+2y^2}{x^2}$$$$\{\begin{array}{l}y=\upsilon x\Rightarrow \frac{dy}{dx}=\upsilon +x\frac{d\upsilon }{dx}\\ \upsilon +x\frac{d\upsilon }{dx}=\frac{3\upsilon x^2+2{\upsilon }^2{x}^2}{x^2}\end{array}$$$$\Rightarrow \upsilon +x\frac{d\upsilon }{dx}=3\upsilon +2{\upsilon }^2$$$$x\frac{d\upsilon }{dx}=2\upsilon +2{\upsilon }^2\to \frac{d\upsilon }{\upsilon (v+1)}=2\frac{dx}{x}$$$$\int (\frac{1}{v}-\frac{1}{v+1})dv=\ln {Cx}^2$$$$\ln \left(\frac{v}{v+1}\right)=\ln {Cx}^2$$$$\frac{y}{y+x}={Cx}^2\Rightarrow y=(y+x){Cx}^2$$$$\left(JS\mathrm{♠}⧫\right)$$

Commented by bubugne last updated on 30/Jul/20

$$\mathrm{correction}$$$$\ln \left(\frac{v}{v+1}\right)=\ln {Cx}^2$$$$\ln \left(\frac{y}{y+x}\right)=\ln {Cx}^2\Rightarrow y=(y+x){Cx}^2$$$$cy-x^{2}y=x^3(c=\frac{1}{C})$$$$\mathit{y}=\frac{{\mathit{x}}^3}{\mathit{c}-{\mathit{x}}^2}$$$$$$$$\mathrm{verification}:$$$$x^2\frac{dy}{dx}-3xy-2y^2=x^2\frac{3{cx}^2-3x^4+2x^4}{{(c-x^2)}^2}-\frac{3x^4}{c-x^2}-\frac{2x^6}{{(c-x^2)}^2}$$$$x^2\frac{dy}{dx}-3xy-2y^2=\frac{3{cx}^4-x^6}{{(c-x^2)}^2}-\frac{3{cx}^4-3x^6}{{(c-x^2)}^2}-\frac{2x^6}{{(c-x^2)}^2}$$$$x^2\frac{dy}{dx}-3xy-2y^2=\frac{3{cx}^4-x^6-3{cx}^4+3x^6-2x^6}{{(c-x^2)}^2}$$$$x^2\frac{dy}{dx}-3xy-2y^2=0$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com