The integrating factor of the differential equation \(x\frac{dy}{dx}-y=2x^{2}\) is
The integrating factor of the differential equation \(x\frac{dy}{dx}-y=2x^{2}\) is
\(e-x\)
\(e-y\)
\(\frac{1}{x}\)
\(x\)
The Correct Option is C
Solution and Explanation
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.
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Concepts Used:
General Solutions to Differential Equations
A relation between involved variables, which satisfy the given differential equation is called its solution. The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution and the solution free from arbitrary constants is called particular solution.
For example,
Read More: Formation of a Differential Equation