The general solution of the differential equation `(dy)/(dx)+y\ g^(prime)(x)=g(x)g^(prime)(x)` , where `g(x)` is a given function of `x` , is a. `g(x)+log{1+y+g(x)}=C` b. `g(x)+log{1+y-g(x)}=C` c. `g(x)-log{1+y-g(x)}=C` d. None of these

Explanation

`frac{d y}{d x}+y g^{prime}(x)=g(x) g^{prime}(x) ldots . .(1)`

`frac{d y}{d x}+P y=Q `

where ` P=g^{prime}(x) and Q=g(x) g^{prime}(x) . `

`therefore I . F .=e^{int P d x} `

`=e^{int g^{prime}(x) d x} `

`=e^{g(x)}`

Multiplying both I.F.

`e^{g(x)}(frac{d y}{d x}+y g^{prime}(x))=e^{g(x)} g(x) g^{prime}(x) `

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