If f (x) and g (x) are two solutions of the differential equation `(a(d^2y)/(dx^2)+x^2(dy)/(dx)+y=e^x, then `f(x)-g(x)` is the solution of

Options

A. `a^(2)(d^(2)y)/(dx^(2))+(dy)/(dx)+y=e^(x)`
B. `a^(2)(d^(2)y)/(dx^(2))+y=e^(x)`
C. `a(d^(2)y)/(dx^(2))+y=e^(x)`
D. `a(d^(2)y)/(dx^(2))+x^(2)(dy)/(dx)+y=0`

Explanation

To solve the problem, we need to analyze the given differential equation and the implications of having two solutions, \( f(x) \) and \( g(x) \). The main goal is to find the differential equation that \( f(x) - g(x) \) satisfies.

### Step-by-Step Solution:

1. **Identify the Given Differential Equation**:
The given differential equation is:
\[ a \frac{d^2y}{dx^2} + x^2 \frac{dy}{dx} + y = e^x \]
where \( f(x) \) and \( g(x) \) are two solutions.

2. **Substituting the Solutions**:
Since \( f(x) \) and \( g(x) \) are solutions to the differential equation, we can write:
\[ a \frac{d^2f}{dx^2} + x^2 \frac{df}{dx} + f = e^x \quad \text{(1)} \]
\[ a \frac{d^2g}{dx^2} + x^2 \frac{dg}{dx} + g = e^x \quad \text{(2)} \]

3. **Subtract the Two Equations**:
Now, we subtract equation (2) from equation (1):
\[ \left(a \frac{d^2f}{dx^2} - a \frac{d^2g}{dx^2}\right) + \left(x^2 \frac{df}{dx} - x^2 \frac{dg}{dx}\right) + (f - g) = 0 \]
This simplifies to:
\[ a \frac{d^2}{dx^2}(f - g) + x^2 \frac{d}{dx}(f - g) + (f - g) = 0 \]

4. **Let \( y = f - g \)**:
Define \( y = f - g \). Then we can rewrite the equation as:
\[ a \frac{d^2y}{dx^2} + x^2 \frac{dy}{dx} + y = 0 \]

5. **Conclusion**:
The function \( f(x) - g(x) \) satisfies the differential equation:
\[ a \frac{d^2y}{dx^2} + x^2 \frac{dy}{dx} + y = 0 \]

### Final Answer:
Thus, \( f(x) - g(x) \) is the solution of the differential equation:
\[ a \frac{d^2y}{dx^2} + x^2 \frac{dy}{dx} + y = 0 \]