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##### Chapter 1 / chapter 1 Problem 4 (1)
« on: January 16, 2022, 07:34:18 PM »
$uu_{xy}=u_{x}u_{y}$

$(u_{x}u_{y})/uu_{x}=u_{xy}/u_{x}$

divide both side by$uu_{x}$ and get

$u_{y}/u=u_{xy}/u_{x}$

integrate with respect to y

$\ln{u}+f(x)=\ln{u_{x}}+g(x)$ enough to write one function of $x$

let g(x)-f(x)=n(x)

$u=u_{x}\times n(x)$

$u_{x}/u=n(x)$

$\ln{u}=N(x)+m(y)$

$u=N_{1}(x)\times m(y)$ "another $m(x)$"

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##### Chapter 1 / home assignment1 Q3(1),(2),(3)&(4)
« 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 :
$u=u^2+x\times e^{xy}+F(y)+g(x)$

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