the solution of the differential equation ` dy/dx = ax + b` , `a!=0` represents

Options

A. a parabola
B. an ellipse
C. a circle
D. a hyperbola

Answer

To solve the differential equation \( \frac{dy}{dx} = ax + b \) where \( a \neq 0 \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given differential equation:
\[ \frac{dy}{dx} = ax + b \]

### Step 2: Separate Variables
We can separate the variables by multiplying both sides by \( dx \):
\[ dy = (ax + b) \, dx \]

### Step 3: Integrate Both Sides
Next, we integrate both sides. The left side integrates to \( y \), and the right side requires us to integrate \( ax + b \):
\[ \int dy = \int (ax + b) \, dx \]
This gives us:
\[ y = \frac{a}{2}x^2 + bx + C \]
where \( C \) is the constant of integration.

### Step 4: Rearrange the Equation
We can rearrange the equation:
\[ y = \frac{a}{2}x^2 + bx + C \]

### Step 5: Identify the Shape of the Equation
The equation \( y = \frac{a}{2}x^2 + bx + C \) is a quadratic equation in the form \( y = Ax^2 + Bx + C \). Since \( a \neq 0 \), the coefficient of \( x^2 \) is non-zero, indicating that the graph of this equation is a parabola.

### Conclusion
Thus, the solution of the differential equation \( \frac{dy}{dx} = ax + b \) represents a parabola.