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 93707 by i jagooll last updated on 14/May/20

$$\mathrm{y}{\mathrm{y}}^{″}={\left({\mathrm{y}}^{\prime }\right)}^2$$

Commented by i jagooll last updated on 14/May/20

cool man

Commented by john santu last updated on 14/May/20

$$\mathrm{let}\mathrm{v}\left(\mathrm{y}\right)=\frac{\mathrm{dy}}{\mathrm{dx}}$$$$\Rightarrow \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{v}\left(\mathrm{y}\right)\right)=\frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{dy}}.\frac{\mathrm{dy}}{\mathrm{dx}}$$$$\mathrm{v}\left(\mathrm{y}\right)\frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{dy}}=\mathrm{v}{\left(\mathrm{y}\right)}^2$$$$\Rightarrow \mathrm{solve}\mathrm{yv}\left(\mathrm{y}\right)\frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{dy}}=\mathrm{v}{\left(\mathrm{y}\right)}^2$$$$\mathrm{factor}-\mathrm{v}\left(\mathrm{y}\right)\{-\mathrm{y}\frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{dy}}+\mathrm{v}\left(\mathrm{y}\right)\}=0$$$$\mathrm{v}\left(\mathrm{y}\right)\{-\mathrm{y}\frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{dy}}+\mathrm{v}\left(\mathrm{y}\right)\}=0$$$$\{\begin{array}{l}\mathrm{v}\left(\mathrm{y}\right)=0\\ \frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{dy}}=\frac{\mathrm{v}\left(\mathrm{y}\right)}{\mathrm{y}}\end{array}$$$$\mathrm{for}\frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{dy}}=\frac{\mathrm{v}\left(\mathrm{y}\right)}{\mathrm{y}}$$$$\frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{v}\left(\mathrm{y}\right)}=\frac{\mathrm{dy}}{\mathrm{y}}\Rightarrow \int \frac{\mathrm{d}\left(\mathrm{v}\left(\mathrm{y}\right)\right)}{\mathrm{v}\left(\mathrm{y}\right)}=\int \frac{\mathrm{dy}}{\mathrm{y}}$$$$\ln \left(\mathrm{v}\left(\mathrm{y}\right)\right)=\ln \left(\mathrm{y}\right)+{\mathrm{c}}_1$$$$\mathrm{v}\left(\mathrm{y}\right)=\mathrm{Cy}\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{Cy}$$$$\frac{\mathrm{dy}}{\mathrm{y}}=\mathrm{Cx}\Rightarrow \ln \mathrm{y}=\frac{1}{2}{\mathrm{Cx}}^2+{\mathrm{c}}_2$$$$\mathrm{y}={\mathrm{e}}^{\frac{1}{2}{\mathrm{Cx}}^2+{\mathrm{c}}_2}.$$$$\mathrm{for}\mathrm{v}\left(\mathrm{y}\right)=0\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=0$$$$\mathrm{y}={\mathrm{c}}_3\Rightarrow \{\begin{array}{l}\mathrm{y}={\mathrm{e}}^{\frac{1}{2}{\mathrm{Cx}}^2+{\mathrm{c}}_2}\\ \mathrm{y}={\mathrm{c}}_3\end{array}$$$$$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com