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Chapter 1 / chapter 1 Problem 4 (1)
« on: January 16, 2022, 07:34:18 PM »


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


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)

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



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


Chapter 1 / home assignment1 Q3(1),(2),(3)&(4)
« on: January 16, 2022, 04:49:37 PM »
$u_{xy}=0,denote: v=u_{x}$
$u=F(x)+g(y), (let F'(x)=f(x))$

let$ u_{x} = v$, so
$ u_{xy}=v_{y}$
$therefore: v_{y}=v$ integrate on both sides
$v=u_{x}=e^{2y}\times f_2(x)$
let $f_{2}(x)=e^{f_{1}(x)}$
$u=f_{3}(x)\times e^{2y}+g(y)$
where $f'_{3}(x)=f_{2}(x)$


 integrate on both sides
the general solution is :
$u=u^2+x\times e^{xy}+F(y)+g(x)$

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