The complementary differential operator is devised to decouple two coupled partial differential equations with variable coefficients. This complementary operator transforms two coupled partial differential equations into a pair of uncoupled equations either of (i) fourth order, (ii) third order, or (iii) second order. The conditions for the existence of linearly dependent solutions for two coupled equations with constant coefficient is also discussed. The method is applied to the propagation of

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... two sound waves in liquid helium at finite temperatures.