Q.1 (e) If \(u=2(ax+by)^2-k(x^2+y^2)\) and \(a^2+b^2=k\), find \(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}\).
Best answer
\(\frac{\partial u}{\partial x}=4a(ax+by)-2kx\Rightarrow\frac{\partial^2u}{\partial x^2}=4a^2-2k\)
\(\frac{\partial u}{\partial y}=4b(ax+by)-2ky\Rightarrow\frac{\partial^2u}{\partial y^2}=4b^2-2k\)
\(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=4(a^2+b^2)-4k=4k-4k\)
\(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0\)
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