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\frac{d^{2}y_{0}}{dx^{2}} - \omega^{2}x^{2}y_{0} + \omegay_{0} = 0 (*)
\Rightarrow y_{0}'(x) = -2\lambdaxe^{-\lambda x^{2}} (2)
\Rightarrow y_{0}''(x) = -2\lambdae^{-\lambda x^{2}} + 4\lambda^{2}x^{2}e^{-\lambda x^{2}}
-2\lambdae^{-\lambda x^{2}} + 4\lambda^{2}x^{2}e^{-\lambda x^{2}} -\omega^{2}x^{2}e^{-\lambda x^{2}} + \omegae^{-\lambda x^{2}} = 0 (**)
\Rightarrow \lamda = \frac{\omega}{2}
y_{0}'(x) = -\omegaxe^{-\frac{\omega}{2} x^{2}}
\displaystyle x=1+y \\ \implies dx=dy \\ x^2=1+2y+y^2 \\ \\ \\ I=\int_0^1\frac{(2-2(1+y)+1+2y+y^2)^k}{(1+y)^{k+1}}dy \\ \\[br]=\int_0^1\frac{(1+y^2)^k}{(1+y)^{k+1}}dy
[br]f'(x) = 2ax + b \\[br]f(1) = a + b + c \\[br]f(-1) = a - b + c \\[br]f(0) = c[br]
x^2 + bk + c = 0 \\[br]\dfrac{1}{k^2} + \dfrac{b}{k} + c = 0 \implies 1 + bk + ck^2 = 0 \implies k^2 + \dfrac{b}{c}k + \dfrac{1}{c} = 0
\alpha^3 + p\alpha^2 + q\alpha + r = 0 \\[br](1-\alpha)^3 + p(1 - \alpha)^2 + q(1-\alpha) + r = 0 \\[br]\dfrac{1}{\alpha^3} + \dfrac{p}{\alpha^2} + \dfrac{q}{\alpha} + r = 0
1 - 3\alpha + 3\alpha^2 - \alpha^3 + p - 2\alpha p + p\alpha^2 + p - p\alpha + 1 = 0 \\[br]\implies \alpha^3 - 1 + 3\alpha - 3\alpha^2 - p + 2\alpha p - p \alpha^2 - p + p\alpha - 1 = 0 \\[br]\implies \alpha^3 + (-3-p)\alpha^2 + (3+3p)\alpha - (2p + 2) = 0
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