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Found problems: 1

1942 Putnam, B3
Tags: partial differential equations , differential equation
Given $x=\phi(u,v)$ and $y=\psi(u,v)$, where $ \phi$ and $\psi$ are solutions of the partial differential equation $$(1) \;\,\;\, \; \frac{ \partial \phi}{\partial u} \frac{\partial \psi}{ \partial v} - \frac{ \partial \phi}{\partial v} \frac{\partial \psi}{ \partial u}=1.$$ By assuming that $x$ and $y$ are the independent variables, show that $(1)$ may be transformed to $$(2) \;\,\;\, \; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}.$$ Integrate $(2)$ and show how this effects in general the solution of $(1)$. What other solutions does $(1)$ possess?