Question

Let \( f(x) \) be a positive function such that the area bounded by \( y = f(x) \), \( y = 0 \), from \( x = 0 \) to \( x = a>0 \) is \[ \int_0^a f(x) \, dx = e^{-a} + 4a^2 + a - 1. \] Then the differential equation, whose general solution is \[ y = c_1 f(x) + c_2, \] where \( c_1 \) and \( c_2 \) are arbitrary constants, is:

• A. $(8e^x - 1)\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$
• B. $(8e^x + 1)\frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$
• C. $(8e^x + 1)\frac{d^2y}{dx^2} + \frac{dy}{dx} = 0$
• D. $(8e^x - 1)\frac{d^2y}{dx^2} - \frac{dy}{dx} = 0$

Answer 1

The Correct Option is C

To determine the differential equation with the general solution \( y = c_1 f(x) + c_2 \), where \( c_1 \) and \( c_2 \) are arbitrary constants, we are provided with the area under the curve \( y = f(x) \) from \( x = 0 \) to \( x = a\) as \(\int_0^a f(x) \, dx = e^{-a} + 4a^2 + a - 1\).

The structure of the general solution \( y = c_1 f(x) + c_2 \) implies that \( f(x) \) is a particular solution to the associated homogeneous differential equation, and \( c_2 \) represents a constant solution.

The process is as follows:

First, differentiate the general solution \( y = c_1 f(x) + c_2 \) with respect to \( x \):

1. \(\frac{dy}{dx} = c_1 \frac{df}{dx}\)

Differentiate again to obtain the second derivative:

1. \(\frac{d^2y}{dx^2} = c_1 \frac{d^2f}{dx^2}\)

The differential equation is expected to be linear and of the second order. We will substitute the expressions for \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) into the provided options.

Next, we examine each option by substituting \( y = f(x) \) into the differential equation. The option for which this substitution results in the equation holding true identifies \( f(x) \) as a solution.

• Option 1: \((8e^x - 1)\frac{d^2 y}{dx^2} + \frac{dy}{dx} = 0\)
• Option 2: \((8e^x + 1)\frac{d^2 y}{dx^2} - \frac{dy}{dx} = 0\)
• Option 3: \((8e^x + 1)\frac{d^2 y}{dx^2} + \frac{dy}{dx} = 0\)
• Option 4: \((8e^x - 1)\frac{d^2 y}{dx^2} - \frac{dy}{dx} = 0\)

After evaluating these expressions and utilizing the derived form of \( f(x) \) from the given integral, we find that:

• Option 3 is the correct choice because substituting \( f(x) \) into this equation simplifies the entire expression to zero, confirming that \( f(x) \) is a solution.

Therefore, the differential equation that yields the specified general solution is:

1. \((8e^x + 1)\frac{d^2 y}{dx^2} + \frac{dy}{dx} = 0\)

This conclusion aligns with the preceding analysis, validating Option 3 as the correct differential equation.