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 integrate both sides step by step.

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

### Step 2: Separate Variables
We can rewrite this as:
\[ dy = (ax + b)dx \]

### Step 3: Integrate Both Sides
Now, we will integrate both sides. The left side integrates to \(y\) and the right side requires us to integrate \(ax + b\):
\[ \int dy = \int (ax + b)dx \]

### Step 4: Perform the Integration
The integration of the left side gives:
\[ y = \int dy = y \]
For the right side, we integrate:
\[ \int (ax + b)dx = \frac{a}{2}x^2 + bx + C \]
where \(C\) is the constant of integration.

### Step 5: Combine the Results
Combining both results, we have:
\[ y = \frac{a}{2}x^2 + bx + C \]

### Step 6: Identify the Type of Function
The equation \(y = \frac{a}{2}x^2 + bx + C\) is a quadratic function in \(x\), which represents a parabola.

### Conclusion
Thus, the solution of the differential equation \(\frac{dy}{dx} = ax + b\) where \(a \neq 0\) represents a parabola.