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Consider the initial value problem for a first-order differential equation  \[    y ′ = f ( x , y ) , y ( x 0 ) = y 0 . y’ = f(x,y),\quad y({x_0}) = {y_0}.     \]  In this paper a new uniqueness criterion is proved. This criterion is related to the numeric equation  \[    u = y 0 + ( t −  x 0 ) f ( t , u ) . u = {y_0} + (t - {x_0})f(t,u).     \]  It is also shown that some well-known uniqueness theorems are consequences of our result.

A uniqueness theorem for 𝑦’=𝑓(𝑥,𝑦),𝑦(𝑥₀)=𝑦₀