Question

The integrating factor of the differential equation \(x\frac{dy}{dx}-y=2x^{2}\) is

• A. \(e-x\)
• B. \(e-y\)
• C. \(\frac{1}{x}\)
• D. \(x\)

Answer 1

The Correct Option is C

The given differential equation is:

\(x\frac{dy}{dx}-y=2x^{2}\)

\(⇒\frac{dy}{dx}-\frac{y}{x}=2x\)

This is a linear differential equation of the form:

\(\frac{dy}{dx}+py=Q\)\(=Q(where\) \(p=\)\(\frac{-1}{x}\) \(and\) \(Q=2x\)).

The integrating factor(I.F.)is given by the relation,

\(e^{∫pdx}\)

\(∴I.F.=\) \(e^{∫\frac{-1}{x}dx}\)=\(e^{-logx}\)=\(e^{log(x^{-1})}\)=\(x^{-1}\)=\(\frac{1}{x}\)

Hence,the correct answer is C.