Summary: The solution of differential equations lies at the heart of many problems in structural economics. In econometrics the general nonparametric analysis of consumer welfare is historically the most obvious application, but there are also many applications in finance and other fields. This work considers the general nonparametric form for these problems and identification conditions. It derives a kernel-based estimator and shows consistency and asymptotic normality. In particular, the link with inverse problems allows us to define it in terms of a well-posed inverse problem and to stress the regularity properties of the estimated solutions.