Variation of parameters

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Theorem: Let $t_0 \in \mathbb{T}^{\kappa}$. Suppose that $y_1$ and $y_2$ form a fundamental set of solutions of the homogeneous equation $L_2y=0$. Then the solution of the initial value problem $$L_2y(t)=g(t), y(t_0)=y_0, y^{\Delta}(t_0)=y_0^{\Delta}$$ is given by $$y(t) = \alpha_0y_1(t) + \beta_0y_2(t) + \displaystyle\int_{t_0}^t \dfrac{y_1(\sigma(\tau))y_2(t)-y_2(\sigma(\tau))y_1(t)}{W(y_1,y_2)(\sigma(\tau))} g(\tau) \Delta \tau,$$ where the constants $\alpha_0$ and $\beta_0$ are given by $$\alpha_0 = \dfrac{y_2^{\Delta}(t_0)y_0-y_2(t_0)y^{\Delta}_0}{W(y_1,y_2)(t_0)}$$ and $$\beta_0=\dfrac{y_1(t_0)y_0^{\Delta}-y_1^{\Delta}(t_0)y_0}{W(y_1,y_2)(t_0)}.$$

Proof: proof goes here █