Abstract: Conditions of existence of nontrivial isentropic solutions to first order conservation laws are found. Applications are given to the problem on functional dependence for partial derivatives of a smooth function of two variables. In particular, we find conditions on the function $\varphi$, which are necessary for existence of nontrivial $C^1$-smooth solutions for the equation $\frac{\partial v}{\partial t}=\varphi(\frac{\partial v}{\partial x})$.
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