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Mathematics
Differential Equations
Solve the Partial differential equation ##U_{xy}=0##
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[QUOTE="chwala, post: 6828264, member: 287397"] [B]TL;DR Summary:[/B] I am going through these notes; i want to check that i am gettting it right... Solve the pde; ##U_{xy}=0## This is part of the notes; [ATTACH type="full" width="1043px" alt="1670243380258.png"]318230[/ATTACH] My own way of thought; Given; ##U_{xy}=0## then considering ##U_x## as on ode in the ##y## variable; we integrate both sides with respect to ##y## i.e ##\dfrac{du}{dx} \int \dfrac{1}{dy} dy=\int 0 dy## this is the part i need insight...the original problem involves partial derivatives but in this case when we integrate with respect to ##y## we are integrating with respect to ##dy## and not ##∂y## ...correct? ##U_x= 0 + k##, where ##k## is a constant in terms of ##x## therefore, ##U_x= f(x)##, an arbitrary function of ##x##... which is an ode for ##u## in the ##x## variable. On integrating again with respect to ##x## we get,##U(x,y) = F(x) + k##, where ##k## is a constant in terms of ##y## therefore, ##U(x,y) = F(x) + H(y)##cheers! [/QUOTE]
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Mathematics
Differential Equations
Solve the Partial differential equation ##U_{xy}=0##
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