Question

Find the integrating factor for the following differential equation: $straight x space log space straight x dy over dx space plus space straight y space equals space 2 logx space$

Solution

Consider the given differential equation:
$xlogx dy over dx plus straight y space equals space 2 logx$
Dividing the above equation by xlogx, we have,
$space space fraction numerator xlogx space over denominator xlogx end fraction dy over dx space plus space fraction numerator straight y over denominator straight x space logx end fraction space equals space fraction numerator 2 logx over denominator xlogx end fraction rightwards double arrow dy over dx plus Py space equals space straight Q comma space where space straight P space and space straight Q space are space functions space of space straight x Comparing space equation space left parenthesis 1 right parenthesis space and space the space general space equation comma space we space have comma space straight P left parenthesis straight x right parenthesis space equals space fraction numerator 1 over denominator straight x space logx end fraction space and space straight Q left parenthesis straight x right parenthesis space equals space 2 over straight x The space intergrating space factor space is space given space by space the space formulae space straight e to the power of integral Pdx end exponent space Thus comma space straight I. straight F. space equals space straight e to the power of integral Pdx end exponent space space equals space straight e to the power of integral fraction numerator dx over denominator straight x space log space straight x end fraction end exponent$
$Consider space straight I space equals space integral dx over xlogx Substituting space logx equals space straight t semicolon space space dx over straight x space equals space dt Thus space straight I space equals space integral dt over straight t space equals space log left parenthesis straight t right parenthesis space equals space log space left parenthesis log space straight x right parenthesis Hence comma space straight I. straight F. equals space straight e to the power of integral fraction numerator dx over denominator straight x space log space straight x end fraction end exponent space equals space straight e to the power of log left parenthesis log space straight x right parenthesis end exponent space equals space log space straight x$

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Integrals

Find the integrating factor for the following differential equation: $straight x space log space straight x dy over dx space plus space straight y space equals space 2 logx space$

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