The solution of the differential equation `(dy)/(dx)-(y(x+1))/x=0` is given by a. `y=x e^(x+C)` b. `x=y e^x` c. `y=x+C` d. `x y=e^x+C`

Explanation

`frac{d y}{d x}-frac{y(x+1)}{x}=0 `

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

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

Integrating

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

`Rightarrow int frac{d y}{y}=int d x+int frac{1}{x} d x `

`Rightarrow log y=x+log x+C `

`Rightarrow log y-log x=x+C `

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