The solution of the differential equation `(dy)/(dx)=y/x+(varphi(y/x))/(varphi^(prime)(y/x))` is a. `varphi(y/x)=k x` b. `x \ varphi\ (y/x)=k` c. `varphi\ (y/x)=k y` d. `y \ varphi(y/x)=k`

Explanation

a is correct answer


Use `y=v x`


`

frac{d y}{d x}=v+x cdot frac{d v}{d x} Rightarrow v+x cdot frac{d v}{d x}=v+frac{phi(v)}{phi^{prime}(v)} `

`
Rightarrow x cdot frac{d v}{d x}=frac{phi(v)}{phi^{prime}(v)} `

`
Rightarrow int frac{phi^{prime}(v) d v}{phi(v)}=int frac{d x}{x}`



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