y^m+y^-m=2x হলে m²(x²-1)y₂+m².xy₁-y=? 

$\mathrm{y}^{\mathrm{m}}+\mathrm{y}^{-\mathrm{m}}=2 \mathrm{x} \Rightarrow m \mathrm{y}^{\mathrm{m}-1} \mathrm{y}_{1}-\mathrm{my} \mathrm{y}^{-\mathrm{m}-1}=2$

$\begin{array}{l} \Rightarrow y_{1}=\frac{2}{m y^{m-1}-m y^{-m-1}} \\ \Rightarrow m y_{1}=\frac{2}{y^{m-1}-y^{-m-1}} \\ \Rightarrow m y_{1}=\frac{2 y}{y^{m}-y^{-m}} \ldots \ldots . . . (i)\end{array}$

$\begin{array}{l} \Rightarrow m y_{1}\left(y^{m}-y^{-m}\right)=2 y \\ \Rightarrow m y_{2}\left(y^{m}-y^{-m}\right)+m y_{1}\left\{m y^{m-1} \cdot y_{1}+m y^{-m-1} y_{1}\right\}=2 y_{1} \\ \Rightarrow m y_{2}\left(y^{m}-y^{-m}\right)+m^{2} y_{1}^{2}\left\{y^{m-1}+y^{-m-1}\right\}=2 y_{1} \\ \Rightarrow m y_{2}\left(y^{m}-y^{-m}\right)+m^{2} y_{1}{ }^{2}\left\{y^{m}+y^{-m}\right\}+\frac{1}{y}=2 y_{1} \\ \Rightarrow m y_{2} \frac{2 y}{m y_{1}}+m^{2} y_{1}^{2} \cdot 2 x \frac{1}{y}=2 y_{1} \\ \Rightarrow y_{2} \cdot \frac{y^{2}}{y_{1}^{2}}+m^{2} x y_{1}-y=0 \ldots \ldots \text { (ii) [উভয়পক্ষে } \frac{y_{1}}{y_{1}} \text { করে] } \\ \Rightarrow y_{2}\left[\left(\frac{y^{m}-y^{-m}}{2} m\right)\right]^{2}+m^{2} x y_{1}-y=0[(i) \text { নং থেকে] } \\ \Rightarrow y_{2} \frac{\left(y^{m}-y^{-m}\right)^{2}}{4} m^{2}+m^{2} x_{1}-y=0 \\ \Rightarrow y_{2} \frac{\left(y^{m}+y^{-m}\right)^{2}-4}{4} m^{2}+m^{2} x y_{1}-y=0 \\ \Rightarrow y_{2}\left[\left(\frac{y^{m}+y^{-m}}{2}\right)^{2}-1\right] m^{2}+m^{2} x_{1}-y=0 \\ \Rightarrow y_{2}\left(x^{2}-1\right) m^{2}+m^{2} x y_{1}-y=0 \end{array}$