Word Count : 617

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हमारे पास आंशिक अवकल समीकरण है: [ (D^2 - DD' - 2D) z = \sin(3x + 4y) + e^{2x + y}. ] यहाँ (D = \frac{\partial}{\partial x}) और (D' = \frac{\partial}{\partial y}) है। पहले इसे निम्न रूप में लिखते हैं: [ \frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 z}{\partial x \partial y} - 2 \frac{\partial z}{\partial x} = \sin(3x + 4y) + e^{2x + y}. ] हम हल को दो भागों में विभाजित करते हैं: (z = z_h + z_p), जहाँ (z_h) है समतल समीकरण का सामान्य हल और (z_p) है विशेष हल। 1. समतल समीकरण (Homogeneous Equation): [ \frac{\partial^2 z}{\partial x^2} - \frac{\partial^2 ___ _________ ___ ___ _______ ___.
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