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हल कीजिए :
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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 z}{\partial x \partial y} - 2 \frac{\partial z}{\partial x} = 0. ] मानते हैं (z_h = e^{\alpha x + \beta y}).
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Related Questions
Solve: .
प्राचल विचरण विधि द्वारा समीकरण$
का हल प्राप्त कीजिए।
Using the method of undetermined coefficients, solve the equation
आंशिक अवकल समीकरण :$
का समाकल पृष्ठ ज्ञात कीजिए जो वृत्त से गुजरता हो।
Apply the method of variation of parameter to solve the differential equation:
अवकल समीकरण को हल कीजिए। एक कण विरामावस्था से एक माध्यम, जिसमें प्रतिरोध
प्रति इकाई द्रव्यमान है, में नीचे गिरता है। v किसी भी समय t पर कण का वेग है। सिद्ध कीजिए कि कण द्वारा समय t में तय की गई दूरी
है, जहाँ g गुरुत्वीय त्वरण है।
प्राचल विचरण विधि से निम्नलिखित अवकल समीकरण: को हल कीजिए।