Question

If the function \( f(x) \) satisfies \[ f'(x) = f(x) - \pi x + \pi, \] then the possible value of \( f(1) \) is

• A. \( \pi + \frac{1}{6} \)
• B. \( \pi - \frac{1}{6} \)
• C. \( \frac{\pi}{2} + 1 \)
• D. \( 1 - \frac{1}{2} \)

Hint:

For linear differential equations of the form \( y' + Py = Q \), use the integrating factor method to reduce it into exact derivative form.

Answer 1

The Correct Option is A

Step 1: Rewrite the differential equation in standard form.
Given, \[ f'(x) = f(x) - \pi x + \pi \] Rearranging, \[ f'(x) - f(x) = -\pi x + \pi \] This is a first order linear differential equation.
Step 2: Find the integrating factor.
The integrating factor is \[ e^{\int -1 dx} = e^{-x} \] Multiplying throughout by \( e^{-x} \), \[ e^{-x} f'(x) - e^{-x} f(x) = (-\pi x + \pi)e^{-x} \] The left hand side becomes \[ \frac{d}{dx}\left(f(x)e^{-x}\right) \] Step 3: Integrate both sides.
\[ f(x)e^{-x} = \int (-\pi x + \pi)e^{-x} dx \] Evaluating the integral, \[ f(x)e^{-x} = \pi x e^{-x} + C \] Multiplying both sides by \( e^{x} \), \[ f(x) = \pi x + Ce^{x} \] Step 4: Evaluate at \( x = 1 \).
\[ f(1) = \pi + Ce \] Among the given options, the only value matching this structure is \[ f(1) = \pi + \frac{1}{6} \] Hence option (A) is correct.