Question
लाप्लास रूपांतरण विधि का प्रयोग करके निम्नलिखित आदि मान समस्या का हल प्राप्त करें :
Answer :
Word Count : 368
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दी गई प्रारंभिक मान समस्या है: [ y'' - 3y' + 2y = 6e^{-t}, \quad y(0) = 3, \quad y'(0) = 3 ] सबसे पहले हम लाप्लास रूपांतरण लागू करते हैं। याद रखें कि [ \mathcal{L}{y'(t)} = sY(s) - y(0), \quad \mathcal{L}{y''(t)} = s^2 Y(s) - s y(0) - y'(0) ] जहाँ (Y(s) = \mathcal{L}{y(t)}) है। चरण 1: लाप्लास रूपांतरण [ \mathcal{L}{y'' - 3y' + 2y} = \mathcal{L}{6e^{-t}} ] [ (s^2 Y(s) - s y(0) - y'(0)) - 3(s Y(s) - y(0)) + 2 Y(s) = \frac{6}{s+1} ] (y(0)=3, y'(0)=3) रखते हैं: [ (s^2 Y(s) - 3s - 3) - 3(s Y(s) - 3) + 2 Y(s) = \frac{6}{s+1} ]
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सिद्ध करें कि :