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निम्न समीकरण

का सामान्य हल है

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CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)
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  4. z = ϕ(|x| + |y|), ϕ ∈ C1

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व्याख्या:

दिया गया है: अर्थात xp + yq = 0

Pp + Qq = R से तुलना करने पर, हमारे पास है-

P = x, Q = y और R = 0

इसलिए, लैग्रेंज सहायक समीकरण द्वारा

अब dz = 0

⇒ z = c1

पहले और दूसरे पद का उपयोग करके

समाकलन करने पर,

log |x| = log |y| + log c2

इसलिए, सामान्य हल है -

c1 ϕ(c2) या c2 = ϕ(c1) या ϕ(c1 c2) = 0

⇒ z = ϕ

⇒ विकल्प (1) सही है।

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