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Let $\mathrm{y}=\frac{\mathrm{x}^{2}}{(\mathrm{x}+1)^{2}(\mathrm{x}+2)}$. Then $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}$ is
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Verified Answer
The correct answers are:
$2\left[\frac{3}{(x+1)^{4}}-\frac{3}{(x+1)^{3}}+\frac{4}{(x+2)^{3}}\right]$
Hint:
By partial fraction technique
$\begin{array}{l}
y=\frac{x^{2}}{(x+1)^{2}(x+2)}=\frac{4}{(x+2)}-\frac{3}{(x+1)}+\frac{1}{(x+1)^{2}} \\
\Rightarrow y^{\prime \prime}=\frac{6}{(x+1)^{4}}-\frac{6}{(x+1)^{3}}+\frac{8}{(x+2)^{3}}
\end{array}$
By partial fraction technique
$\begin{array}{l}
y=\frac{x^{2}}{(x+1)^{2}(x+2)}=\frac{4}{(x+2)}-\frac{3}{(x+1)}+\frac{1}{(x+1)^{2}} \\
\Rightarrow y^{\prime \prime}=\frac{6}{(x+1)^{4}}-\frac{6}{(x+1)^{3}}+\frac{8}{(x+2)^{3}}
\end{array}$
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