If $(a x^{2} + b x + c) y + a^{'} x^{2} + b^{'} x + c^{'} = 0$ , find the conditions the conditions that may be rational function of $y$ .

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Solution

${(b b^{'} - 2 a c^{'} - 2 a^{'} c)}^{2} = (b^{2} - 4 a c) ({b^{'}}^{2} - 4 a^{'} c^{'})$ which reduces to
${(a^{'} c + a c^{'})}^{2} = (a b^{'} + a^{'} b) (b c^{'} + b^{'} c)$

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