Word Count : 361

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दी गई अवकल समीकरण है: [ y'' - 2y' + y = \frac{12 e^x}{x^3}. ] सबसे पहले, इसका समांतर (homogeneous) समीकरण देखें: [ y'' - 2y' + y = 0. ] समान्य हल के लिए चरित्र समीकरण है: [ r^2 - 2r + 1 = 0 \implies (r-1)^2 = 0 \implies r = 1 \text{ (दोहराव) }. ] इसलिए समान्य हल होगा: [ y_h = (C_1 + C_2 x)e^x. ] अब प्राचल विचरण (variation of parameters) विधि का प्रयोग करें। समान्य हल के दो स्वतंत्र हल हैं: [ y_1 = _______ _______ __________ __________ _____ ___ ___ ________.
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