Word Count : 354

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Let D = ∂/∂x and D′ = ∂/∂y. The given equation is (D² + 5DD′ + 5D′²)z = x sin(3x − 2y). First find the complementary function. Assume z = e^{mx + ny}. Then the auxiliary equation is m² + 5mn + 5n² = 0. Dividing by n² (n ≠ 0), (m/n)² + 5(m/n) + 5 = 0. Let m/n = r. Then r² + 5r + 5 = 0 __________ ___ ___ ___ __________ ________.
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