find the solution of the following differential equation ` x logx (dy)/(dx) + y = 2 log x`

Explanation

`x log x frac{d y}{d x}+y=2 log x`

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


Use `P=frac{1}{x log x} Rightarrow int P d P=int frac{1}{x log x} d x=log (log x) `

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


by Multiplying (1) with I.F. we obtain

`
log x frac{d y}{d x}+frac{y}{x}=frac{2}{x} log x`


by Integrating sides we obtain

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


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