Solve this using fourier transform of sine function
\(\displaystyle v_{xx}\, +\, v_{yy}\, =\, 0\, \mbox{ for }\, x\, >\, 0,\, y\, >\, 0\)
\(\displaystyle v_x (0,\, y)\, -\, hv(0,\, y)\, =\, 0\, \mbox{ for }\, y\, >\, 0\)
\(\displaystyle v(x,\, 0)\, =\, T_0,\, \mbox{ for }\, x\, >\, 0\)
\(\displaystyle v(x,\, y)\, \rightarrow\, 0,\, v_x(x,\, y)\, \rightarrow\, 0\, v_y(x,\, y)\, \rightarrow\, 0\)
\(\displaystyle \mbox{Then }\, x^2\, +\, y^2\, \rightarrow\, \infty\)
\(\displaystyle v_{xx}\, +\, v_{yy}\, =\, 0\, \mbox{ for }\, x\, >\, 0,\, y\, >\, 0\)
\(\displaystyle v_x (0,\, y)\, -\, hv(0,\, y)\, =\, 0\, \mbox{ for }\, y\, >\, 0\)
\(\displaystyle v(x,\, 0)\, =\, T_0,\, \mbox{ for }\, x\, >\, 0\)
\(\displaystyle v(x,\, y)\, \rightarrow\, 0,\, v_x(x,\, y)\, \rightarrow\, 0\, v_y(x,\, y)\, \rightarrow\, 0\)
\(\displaystyle \mbox{Then }\, x^2\, +\, y^2\, \rightarrow\, \infty\)
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