Solve the following differential equation: `(x^3+x^2+x+1)(dy)/(dx)=2x^2+x`

Explanation

`(x^{3}+x^{2}+x+1) frac{d y}{d x}=2 x^{2}+x`

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Rightarrow frac{d y}{d x}=frac{2 x^{2}+x}{x^{3}+x^{2}+x+1}`

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Rightarrow d y=frac{2 x^{2}+x}{(x+1)(x^{2}+1)} d x`


By Integrating, we obtain

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int d y=int{frac{2 x^{2}+x}{(x+1)(x^{2}+1)}} d x`

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Rightarrow y=int{frac{2 x^{2}+x}{(x+1)(x^{2}+1)}} d x`

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Let frac{2 x^{2}+x}{(x+1)(x^{2}+1)}=frac{A}{x+1}+frac{B x+C}{x^{2}+1}`

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Rightarrow 2 x^{2}+x=A x^{2}+A+B x^{2}+B x+C x+C`

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