The differential equation `y"" (dy)/(dx) + x = a` where a is a constant represents

Options

A. A set of straight lines
B. A set of ellipses
C. A set of circles
D. None of the above

Answer

To solve the differential equation \( y \frac{dy}{dx} + x = a \), where \( a \) is a constant, we can follow these steps:

### Step 1: Rearrange the Equation
Start by rearranging the given differential equation:
\[ y \frac{dy}{dx} = a - x \]

### Step 2: Separate Variables
Next, we can separate the variables \( y \) and \( x \):
\[ y \, dy = (a - x) \, dx \]

### Step 3: Integrate Both Sides
Now, we will integrate both sides:
\[ \int y \, dy = \int (a - x) \, dx \]

### Step 4: Perform the Integration
Calculating the integrals, we have:
\[ \frac{y^2}{2} = ax - \frac{x^2}{2} + C \]
where \( C \) is the constant of integration.

### Step 5: Rearranging the Equation
To rearrange this into a standard form, we can multiply through by 2:
\[ y^2 = 2ax - x^2 + 2C \]

### Step 6: Identify the Conic Section
Rearranging further gives us:
\[ x^2 - 2ax + y^2 = 2C \]
This equation represents a conic section. Completing the square for the \( x \) terms:
\[ (x - a)^2 + y^2 = 2C + a^2 \]
This is the equation of a circle centered at \( (a, 0) \) with radius \( \sqrt{2C + a^2} \).

### Conclusion
Thus, the differential equation \( y \frac{dy}{dx} + x = a \) represents a set of circles.

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