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$$x^{2} y^{\prime \prime}+7 x y^{\prime}+8 y=0$$

$$a x^{2} \frac{d^{2} y}{d x^{2}}+b x \frac{d y}{d x}+c y=0$$

$$a=1 \quad b=7 \quad c=8$$

$$a m^{2}+(b-a) m+c=0$$

$$\rightarrow(m+4)(m+2)=0$$

$$\rightarrow m=-2,-4$$

$$y=c_{1} x^{-2}+c_{2} x^{-4} $$

$$9 x^{2} y^{\prime \prime}+3 x y^{\prime}+y=0$$

$$a=9 \quad b=3 \quad c=1$$

$$a m^{2}+(b-a) m+c=0$$

$$9 m^{2}-6 m+1=0$$

$$(3 m-1)(3 m-1)=0$$

$$m_{1}=1 / 3 \quad m_{2}=1 / 3$$

$$y=c_{1} x^{1 / 3}+c_{2} x^{1 / 3} \ln (x)$$

$$x^{2} y^{\prime \prime}-9 x y^{\prime}+28 y=0$$

$$a=1 \quad b=-9 \quad c=28$$

$$a m^{2}+(b-a) m+c=0$$

$$m^{2}-10 m+28=0$$

$$m=5 \pm \sqrt{3} i$$ Complex Root

$$y=x^{5}\left[c_{1} \cos (\sqrt{3} \ln (x))+c_{2} \sin (\sqrt{3} \ln (x)\right]$$

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