Question
Download Solution PDFm का वह मान क्या है जिसके लिए 2x – x2 + my2 हार्मोनिक है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
यदि f(x, y) हार्मोनिक है, तो इसे लाप्लास के समीकरण को संतुष्ट करना चाहिए।
\({{\nabla }^{2}}~f\left( x,~y \right)=0=\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}\)
गणना:
दिया गया फलन: f = 2x – x2 + my2
इसलिए, हार्मोनिक के लिए इसे लाप्लास के समीकरण को संतुष्ट करना चाहिए।
\({{\nabla }^{2}}f=0=\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}\)
\(\frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}=\frac{\partial \left( \frac{\partial f}{\partial x} \right)}{\partial x}=\frac{\partial }{\partial x}~\left( 2-2x \right)=-2\)
\(\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}=\frac{\partial \left( \frac{\partial f}{\partial y} \right)}{\partial y}=\frac{\partial \left( 2my \right)}{\partial y}=2m\)
\(\Rightarrow \frac{{{\partial }^{2}}f}{\partial {{x}^{2}}}+\frac{{{\partial }^{2}}f}{\partial {{y}^{2}}}=-2+2m=0\)
⇒ m = 1Last updated on May 28, 2025
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