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« Last post by **asdfghj ** on* January 16, 2022, 04:49:37 PM* »
(1):

$u_{xy}=0,denote: v=u_{x}$

$u_{xy}=v_{y}=0$

$v=f(x)$

$u=F(x)+g(y), (let F'(x)=f(x))$

(2):

$u_{xy}=2u_{x}$

let$ u_{x} = v$, so

$ u_{xy}=v_{y}$

$therefore: v_{y}=v$ integrate on both sides

$v_{y}/v=2$

$2y+f_{1}(x)=\ln(v)$

$v=u_{x}=e^{2y}\times f_2(x)$

let $f_{2}(x)=e^{f_{1}(x)}$

$u=f_{3}(x)\times e^{2y}+g(y)$

where $f'_{3}(x)=f_{2}(x)$

(3):

$u_{xy}=e^{xy}$

$u_{x}=e^{xy}y+f(x)$

$u(x,y)=e^{xy}xy+F(x)+g(y)$

(4)

$u_{xy}=2u_{x}+e^{x+y}$

$u_{xy}=u_{yx}$

$e^{xy}=D(x,y)$

integrate on both sides

$\int{u_{xy}}=\int{2u_{x}+D(x,y)}$

$u_{y}=2u+xD(x+y)+f(y)$

so

$u=u^2+xD(x,y)+F(y)+g(x)$

the general solution is :

$u=u^2+x\times e^{xy}+F(y)+g(x)$