Question
Download Solution PDFमान लें कि k एक धनात्मक पूर्णांक है। निम्न अवकलन समीकरण पर विचार करें
\(\left\{\begin{aligned} \frac{d y}{d t} &=y^{\frac{5 k}{5 k+2}} \text { for } t>0, \\ y(0) &=0 \end{aligned}\right.\)
निम्न कथनों में से कौन-सा सच है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFअवधारणा :
यदि \(\frac{dy}{dx}=y^n\), n ∈ (0, 1) और y(a) = 0, a ∈ \(\mathbb R\) तब अवकल समीकरण के अनंत स्वतंत्र हल होते हैं।
व्याख्या:
यहाँ \( \frac{d y}{d t}=y^{\frac{5 k}{5 k+2}}\), k ∈ \(\mathbb N\), y(0) = 0
∵ n = \(\frac{5k}{5k+2}\) < 1 ∀ k ∈ \(\mathbb N\)
अतः इसके अनंत हल हैं जो (0, ∞) पर सतत रूप से अवकलनीय हैं।
विकल्प (3) सही है
Last updated on Jul 8, 2025
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