Geodesic
A differential equation without a solution
Matematičeskie zametki, Tome 10 (1971) no. 2, pp. 125-128 
V. V. Grushin. A differential equation without a solution. Matematičeskie zametki, Tome 10 (1971) no. 2, pp. 125-128. http://geodesic.mathdoc.fr/item/MZM_1971_10_2_a0/
@article{MZM_1971_10_2_a0,
     author = {V. V. Grushin},
     title = {A differential equation without a solution},
     journal = {Matemati\v{c}eskie zametki},
     pages = {125--128},
     year = {1971},
     volume = {10},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_10_2_a0/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A direct construction is given of a function $f(x_1, x_2)\in C^\infty$, such that the equation $$ \frac{\partial u}{\partial x_1}+ix_1^{2k-1}\frac{\partial u}{\partial x_2}=f $$ has no solution in any neighborhood of the origin; the function $f$ and all its derivatives vanish for $x_1=0$.