We solve the operator equations $$\sum_{i=0}^{n} \, a_{i}(x) L \left[ x^{n-i} \varphi(x) \right] = 0 \quad \text{and} \quad \sum_{i=0}^{n} \, a_{i}(x) L \left[ x^{n-i} \varphi(x) \right] = A \left[ \varphi(x) \right],$$ where the unknown is the linear operator $L$, and obtain the solutions in three equivalent forms: as infinite-order differential operators with variable coefficients, as "translation" operators and as integral operators.