Under which one of the following condition does the solution of `(dy)/(dx) = (ax + b)/(cy + d)` represent a parabola ?

Options

A. 1) a = 0 , c = 0
B. 2) a = 1 , b = 2 , `c ne 0`
C. 3) a = 0 , `c ne 0 , b ne 0`
D. 4) a = 1 , c = 1

Answer

To determine under which condition the solution of the differential equation \(\frac{dy}{dx} = \frac{ax + b}{cy + d}\) represents a parabola, we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to separate the variables:
\[ (cy + d) dy = (ax + b) dx \]

### Step 2: Integrate Both Sides
Next, we integrate both sides:
\[ \int (cy + d) dy = \int (ax + b) dx \]

### Step 3: Perform the Integrations
Now we perform the integrations:
- The left side:
\[ \int (cy + d) dy = \frac{c}{2}y^2 + dy \]
- The right side:
\[ \int (ax + b) dx = \frac{a}{2}x^2 + bx + k \]
where \(k\) is the constant of integration.

### Step 4: Set the Integrals Equal
Setting the two integrals equal gives us:
\[ \frac{c}{2}y^2 + dy = \frac{a}{2}x^2 + bx + k \]

### Step 5: Rearranging the Equation
Rearranging this equation, we can express it as:
\[ \frac{c}{2}y^2 - \frac{a}{2}x^2 + (d - b)x + (k - dy) = 0 \]

### Step 6: Identify the Conditions for a Parabola
For this equation to represent a parabola, we need to ensure that one of the variables is squared while the other is linear. This leads us to the following conditions:
1. If \(c = 0\) (making the \(y^2\) term disappear), then \(a\) must not be zero for the equation to remain valid.
2. If \(a = 0\) (making the \(x^2\) term disappear), then \(c\) must not be zero.

### Conclusion
Thus, the condition under which the solution represents a parabola is:
- \(a = 0\) and \(c \neq 0\).

### Final Answer
The correct condition is that \(a = 0\) and \(c \neq 0\).

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