MathDB

B3

Part of 1942 Putnam

Problems(1)

Putnam 1942 B3

Source: Putnam 1942

3/1/2022
Given x=ϕ(u,v)x=\phi(u,v) and y=ψ(u,v)y=\psi(u,v), where ϕ \phi and ψ\psi are solutions of the partial differential equation (1)        ∂ϕ∂u∂ψ∂v−∂ϕ∂v∂ψ∂u=1.(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 xx and yy are the independent variables, show that (1)(1) may be transformed to (2)        ∂y∂v=∂u∂x.(2) \;\,\;\, \; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}. Integrate (2)(2) and show how this effects in general the solution of (1)(1). What other solutions does (1)(1) possess?
Putnampartial differential equationsdifferential equation