# Find the integrating factor for the following differential equation: x logx dy/dx+y=2log x - Mathematics

Find the integrating factor for the following differential equation:x logx dy/dx+y=2log x

#### Solution

Consider the given differential equation:

x logx dy/dx+y=2log x

Dividing the above equation by xlogx, we have,

(x logx)/(x logx)dy/dx+y/(x logx)=(2log x)/(x logx)

=>dy/dx+y/(x logx)=1/x ........(1)

Consider the general linear differential equation

dy/dx+Py=Q, where P and Q are functions of x.

Comparing equation (1) and the general equation, we have,

P(x)=1/xlogx and Q(x)=2/x

The integrating factor is given by the formula e^(intPdx)

Thus I.F=e^(intPdx)=e^(intdx/(xlogx))

Consider I=int dx/(xlogx)

Substituting logx=t; dx/x=dt

Thus I=intdt/t=log(t)=log(logx)

Hence  I.F=e^(intdx/(xlogx))=e^(log(logx))=logx

Concept: Solutions of Linear Differential Equation
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