$(x^2-y^2-u^2)\cdot u_x+(2xy)\cdot u_y=2xu$ how can I solve this partial dif. equation. I try to use Langarange methods
This is my solution;
$$\frac {dx}{x^2-y^2-u^2}=\frac {dy}{2xy}=\frac {du}{2xu}= λ$$
$$\frac {dy}{2xy}=\frac {du}{2xu}$$
$$\frac {du}{u}-\frac {dy}{y}=0$$
$$\ln u-\ln y=\ln c_1$$
$$\frac {u}{y}=c_1=w$$
then i know that we now need to find another function. but i am stuck.
$$F(w,v)=0$$
$$w=f(v)$$
can u help me to find $v$ with steps, please? i would appreciate it if you help.
On
I think nearly everyone try to use $dx$ part but we can easly use $$\frac{dy-du}{2xy-2xu}=\frac{dy+du}{2xy+2xu}$$ then $$\frac{dy-du}{y-u}=\frac{dy+du}{y+u}$$ and $$\frac{d(y-u)}{y-u}=\frac{d(y+u)}{y+u}$$ $$ln|y-u|=ln|y+u|+ln(d)$$ then $d'=e^{\ln(\frac{y-u}{y+u})}=v$ and $f(v)=w \Rightarrow f(\frac{y-u}{y+u})=\frac{u}{y}$ and $F(w,v)=F(\frac{u}{y},\frac{y-u}{y+u})=0$
There is a another way and this $$\frac{du}{2xu}=\frac{xdx+ydy+udu}{x.(x^2-y^2-u^2)+y.2xy+u.2xu}$$ $$\frac{du}{2xu}=\frac{xdx+ydy+udu}{x^3-x.y^2-x.u^2+2xy^2+2xu^2}$$ $$\frac{du}{2xu}=\frac{xdx+ydy+udu}{x^3+x.y^2+x.u^2}$$ and if we simplify the equation $$\frac{du}{2x.(u)}=\frac{xdx+ydy+udu}{x(x^2+y^2+u^2)}$$ $$\frac{du}{2x.(u)}=\frac{2xdx+2ydy+2udu}{2x(x^2+y^2+u^2)}$$ and cancled the $2x$ and use $d(x^2+y^2+u^2)=2dx+2dy+2du$ then we get $$\frac{du}{u}=\frac{d(x^2+y^2+z^2)}{x^2+y^2+u^2}$$ and $$ln|u|=ln|x^2+y^2+u^2|+lnd$$ $$ln|\frac{u}{x^2+y^2+u^2}|=lnd$$ then $$d=\frac{u}{x^2+y^2+u^2}=v$$ so $F(w,v)=F(\frac{u}{y},\frac{u}{x^2+y^2+u^2})=0$