Question

The integrating factor of the linear differential equation in \(x\) given by \[ \frac{dy}{dx}=\frac{1}{3x+y+2} \] is

• A. \(e^{-3x}\)
• B. \(e^{-x}\)
• C. \(e^{-3y}\)
• D. \(e^{-y}\)

Hint:

If the given differential equation is not linear in the required variable, first invert it. For \[ \frac{dx}{dy}+Px=Q, \] the integrating factor is \[ \boxed{e^{\int P\,dy}}. \]

Answer 1

The Correct Option is C

Step 1: Write the equation in linear form. Given, \[ \frac{dy}{dx} = \frac{1}{3x+y+2}. \] Invert the equation to obtain a linear differential equation in \(x\): \[ \frac{dx}{dy} = 3x+y+2. \] Hence, \[ \frac{dx}{dy}-3x=y+2. \]

Step 2: Find the integrating factor. Comparing with \[ \frac{dx}{dy}+Px=Q, \] we have \[ P=-3. \] Therefore, the integrating factor is \[ \mathrm{I.F.} = e^{\int -3\,dy} = e^{-3y}. \]

Step 3: Write the answer. Hence, \[ \boxed{e^{-3y}}. \] Thus, \[ \boxed{(C)} \] is the correct answer.