Can solution of any differential equation be expressed as power series? If solution of a differential equation can't be expressed as power series, is there any example of it?
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It might depend on what exact definition you have in mind for "differential equation".
If you have a broad enough definition, you can probably construct many counter-examples: take any smooth function which is not the sum of its Taylor series, compute its derivative, construct a differential equation using it.
For example, it is well-known that function $t \mapsto e^{-1/t^2}$ is smooth on $\mathbb{R}$ but that all its derivatives vanish at $t = 0$ (so its Taylor series is null at $t = 0$, and thus the function is not a power series at $0$). However, you can check that it satisfies the following first-order linear homogeneous differential equation: $$ 2y + t^3 y' = 0.$$
