# Abel's theorem

From timescalewiki

Let $\mathbb{T}$ be a time scale and consider the dynamic equation defined by the linear operator $$L_2 y(t) = y^{\Delta \Delta}(t) + p(t) y^{\Delta}(t) + q(t)y(t),$$ where $p,q$ are rd-continuous.

**Theorem:** Let $t_0 \in \mathbb{T}^{\kappa}$ and assume $L_2 y = 0$ is regressive. Suppose that $y_1$ and $y_2$ are two solutions of $L_2 y=0$. Then their wronskian satisfies
$$W(t) = e_{-p+\mu q}(t,t_0)W(t_0)$$
for $t \in \mathbb{T}^{\kappa}$.

**Proof:** █