# PDE , n-variables

#### idontknow

(1) $$\displaystyle \dfrac{\partial Z}{\partial t_1 }+\dfrac{\partial Z}{\partial t_{1}^{2}} +\dotsc + \dfrac{\partial Z}{\partial t_{n}^{n}}=0$$.

(2) $$\displaystyle \partial_x f(x,y) = \partial_y f(x,y)$$.

(3) $$\displaystyle \dfrac{\partial^2 z(x,y)}{\partial x \partial y }=\dfrac{\partial z}{\partial x} + \dfrac{\partial z}{\partial y }$$.

#### romsek

Math Team
You need to look at (1). The way you have it written can't be correct.

(2) is trivially solved by any function $f(x,y) \ni f(x,y)=f(y,x)$

idontknow

#### idontknow

You need to look at (1). The way you have it written can't be correct.

(2) is trivially solved by any function $f(x,y) \ni f(x,y)=f(y,x)$
$$\displaystyle \partial_x f(x,y) = \partial_y f(x,y) \; \Rightarrow x\equiv y \;$$ ; $$\displaystyle \; f(x,y)=f(x,x)\pm f(y,y)=g(x\pm y)$$.

$$\displaystyle f(x,y)=x\pm y$$ satisfies $$\displaystyle [x\pm y]_{x}'=1=[x\pm y]_{y}'=1$$.

can we say the general solution is $$\displaystyle z=Cf(x\pm y)$$ ?

$$\displaystyle \dfrac{\partial }{\partial x} Cf(x\pm y) = \dfrac{\partial }{\partial y } Cf(x\pm y)$$.

For (3) I got : $$\displaystyle y\dfrac{\partial z}{\partial x} +z=0$$.

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