# Solving an equation containing Laplace transform

Consider the equation $$$$\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),$$$$ where $$\mathcal{L}$$ is the Laplace transform of $$y$$ which we assume well defined and $$f,g$$ are two functions such that $$\begin{equation*} f(p)=0\text{ }\Longleftrightarrow p=s_{2},\text{ }g(p)=0\text{ }% \Longleftrightarrow p=s_{1}. \end{equation*}$$ I want to prove that such a function $$y$$ exists if and only if $$p=s_{1}$$ or $$% p=s_{2}.$$ The sufficient part is clear; for $$p=s_{1}$$ we get $$\begin{eqnarray*} \frac{f(s_{1})}{f(s_{1})}\mathcal{L}(y)(s_{1})+0 &=&\mathcal{L(}y)\mathbf{(}% s_{1}), \\ &\Longrightarrow &\mathcal{L}(y)(s_{1})=\mathcal{L}(y)(s_{1}). \end{eqnarray*}$$

The same thing holds for $$p=s_{2}.$$ I want to prove that a solution $$y$$ exists if and only if $$p=s_{1}$$ or $$p=s_{2}.$$ The sufficient part is clear, but I cannot prove the necessary part. Any ideas?. Thank you.