Math, asked by vishwam71, 2 months ago

if u=x³y³/x³+y³,prove that x du/dx + y du/dy=3u​

Answers

Answered by MaheswariS
3

\textbf{Given:}

\mathsf{u=\dfrac{x^3y^3}{x^3+y^3}}

\textbf{To prove:}

\mathsf{x\;\dfrac{\partial\,u}{\partial\,x}+y\;\dfrac{\partial\,u}{\partial\,y}=3\;u}

\textbf{Solution:}

\textbf{Euler's theorem:}

\textsf{If f is  homogeneous function of degree n, then}

\mathsf{x\;\dfrac{\partial\,f}{\partial\,x}+y\;\dfrac{\partial\,f}{\partial\,y}=3\;f}

\textsf{Consider,}

\mathsf{u(x,y)=\dfrac{x^3y^3}{x^3+y^3}}

\mathsf{u(tx,ty)=\dfrac{t^3x^3{\times}t^3y^3}{t^3x^3+t^3y^3}}

\mathsf{u(tx,ty)=\dfrac{t^6x^3y^3}{t^3(x^3+y^3)}}

\mathsf{u(tx,ty)=t^3\left(\dfrac{x^3y^3}{x^3+y^3}\right)}

\mathsf{u(tx,ty)=t^3\;u(x,y)}

\therefore\textsf{u is a homogeneous function of degree 3}

\textsf{By Euler's theorem}

\mathsf{x\;\dfrac{\partial\,u}{\partial\,x}+y\;\dfrac{\partial\,u}{\partial\,y}=n\;u}

\implies\boxed{\mathsf{x\;\dfrac{\partial\,u}{\partial\,x}+y\;\dfrac{\partial\,u}{\partial\,y}=3\;u}}

\textbf{Find more:}

If u=sin^-1(x2+y2/x+y)prove that xdu/dx)+ydu/dy)=tan u

https://brainly.in/question/12684777

Answered by mahek77777
14

\textbf{Given:}

\mathsf{u=\dfrac{x^3y^3}{x^3+y^3}}

\textbf{To prove:}

\mathsf{x\;\dfrac{\partial\,u}{\partial\,x}+y\;\dfrac{\partial\,u}{\partial\,y}=3\;u}

\textbf{Solution:}

\textbf{Euler's theorem:}

\textsf{If f is  homogeneous function of degree n, then}

\mathsf{x\;\dfrac{\partial\,f}{\partial\,x}+y\;\dfrac{\partial\,f}{\partial\,y}=3\;f}

\textsf{Consider,}

\mathsf{u(x,y)=\dfrac{x^3y^3}{x^3+y^3}}

\mathsf{u(tx,ty)=\dfrac{t^3x^3{\times}t^3y^3}{t^3x^3+t^3y^3}}

\mathsf{u(tx,ty)=\dfrac{t^6x^3y^3}{t^3(x^3+y^3)}}

\mathsf{u(tx,ty)=t^3\left(\dfrac{x^3y^3}{x^3+y^3}\right)}

\mathsf{u(tx,ty)=t^3\;u(x,y)}

\therefore\textsf{u is a homogeneous function of degree 3}

\textsf{By Euler's theorem}

\mathsf{x\;\dfrac{\partial\,u}{\partial\,x}+y\;\dfrac{\partial\,u}{\partial\,y}=n\;u}

\implies\boxed{\mathsf\red{x\;\dfrac{\partial\,u}{\partial\,x}+y\;\dfrac{\partial\,u}{\partial\,y}=3\;u}}

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