Question

Which one of the following curves correctly represents (schematically) the solution for the equation $\dfrac{df}{dx}+2f=3;\; f(0)=0$ ?

• A. (A)
• B. (B)
• C. (C)
• D. (D)

Hint:

Linear differential equations of the form $f'+kf=C$ yield exponential saturation curves approaching $\dfrac{C}{k}$.

Answer 1

The Correct Option is B

Step 1: Solve the differential equation.
The equation $\displaystyle \frac{df}{dx}+2f=3$ is a first-order linear DE. Using the integrating factor $e^{2x}$, the solution is $\displaystyle f(x)=\frac{3}{2}\left(1-e^{-2x}\right).$

Step 2: Apply the initial condition.
$f(0)=\dfrac{3}{2}(1-1)=0$, which agrees.

Step 3: Analyze behaviour of solution.
As $x\to\infty$, $\displaystyle f(x)\to \frac{3}{2}.$ Thus the curve starts at \(0\) and asymptotically approaches \(3/2\).

Step 4: Match with the options.
Only option (B) shows a saturation at \(3/2\).