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 145300 by imjagoll last updated on 04/Jul/21

$${\mathrm{yy}}^{\prime }=\mathrm{x}{\mathrm{e}}^{\frac{\mathrm{x}}{\mathrm{y}}}$$

Answered by puissant last updated on 04/Jul/21

$${\mathrm{y}}^{\prime }=\frac{\mathrm{x}}{\mathrm{y}}{\mathrm{e}}^{\frac{\mathrm{x}}{\mathrm{y}}}\mathrm{put}\mathrm{t}=\frac{\mathrm{x}}{\mathrm{y}}$$$$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dt}}={\mathrm{te}}^{\mathrm{t}}\Rightarrow \mathrm{dy}={\mathrm{te}}^{\mathrm{t}}\mathrm{dt}$$$$\Rightarrow \int \mathrm{dy}=\int {\mathrm{te}}^{\mathrm{t}}\mathrm{dt}$$$$\int {\mathrm{te}}^{\mathrm{t}}\mathrm{dt}={\mathrm{te}}^{\mathrm{t}}-{\mathrm{e}}^{\mathrm{t}}+\mathrm{k}={\mathrm{e}}^{\mathrm{t}}(\mathrm{t}-1)+\mathrm{k}$$$$\Rightarrow \mathrm{y}={\mathrm{e}}^{\mathrm{t}}(\mathrm{t}-1)+\gamma ....$$

Commented by imjagoll last updated on 04/Jul/21

$$\mathrm{if}\mathrm{t}=\frac{\mathrm{x}}{\mathrm{y}}\Rightarrow \mathrm{y}=\frac{\mathrm{x}}{\mathrm{t}}$$$$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\mathrm{t}}-\frac{\mathrm{x}}{{\mathrm{t}}^2}\frac{\mathrm{dt}}{\mathrm{dx}}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com