The solution of `(dy)/(dx)=(a x+h)/(b y+k)` represent a parabola when

Options

A. `a=0, b=0`
B. `a=1, b=2`
C. `a=0, b ne 0`
D. `a=2, b=1`

Answer

To determine when the solution of the differential equation \(\frac{dy}{dx} = \frac{ax + h}{by + k}\) represents a parabola, we can follow these steps:

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

### Step 2: Cross Multiply
Cross multiplying gives us:
\[ (by + k) dy = (ax + h) dx \]

### Step 3: Integrate Both Sides
Now we integrate both sides. The left side can be integrated as follows:
\[ \int (by + k) dy = \frac{b}{2} y^2 + ky \]
The right side can be integrated as:
\[ \int (ax + h) dx = \frac{a}{2} x^2 + hx + C \]
where \(C\) is the constant of integration.

### Step 4: Set Up the Integrated Equation
After integrating, we have:
\[ \frac{b}{2} y^2 + ky = \frac{a}{2} x^2 + hx + C \]

### Step 5: Rearranging the Equation
Rearranging the equation gives:
\[ \frac{b}{2} y^2 - \frac{a}{2} x^2 + ky - hx - C = 0 \]

### Step 6: Identify the Conditions for a Parabola
For the above equation to represent a parabola, we need to analyze the coefficients:
- If \(b \neq 0\) and \(a = 0\), the equation simplifies to:
\[ \frac{b}{2} y^2 + ky - C = hx \]
This represents a parabola that opens along the y-axis.
- If \(a \neq 0\) and \(b = 0\), the equation simplifies to:
\[ \frac{a}{2} x^2 + hx - C = ky \]
This represents a parabola that opens along the x-axis.

### Conclusion
Thus, the solution of the differential equation represents a parabola when:
1. \(b \neq 0\) and \(a = 0\) (parabola centered at the y-axis)
2. \(a \neq 0\) and \(b = 0\) (parabola centered at the x-axis)