If `x^(2)y^(3)=(x+y)^(n+2)` and `(dy)/(dx)=y/x` then `n=`

Options

A. 2
B. 3
C. 6
D. 4

Answer

To solve the problem, we start with the given equation and the derivative condition:

1. **Given Equations**:
\[ x^2 y^3 = (x + y)^{n + 2} \]
\[ \frac{dy}{dx} = \frac{y}{x} \]

2. **Differentiate Both Sides**:
We will differentiate the left-hand side using the product rule and the right-hand side using the chain rule.

- **Left-hand Side**:
\[ \frac{d}{dx}(x^2 y^3) = 2xy^3 + 3x^2y^2 \frac{dy}{dx} \]

- **Right-hand Side**:
\[ \frac{d}{dx}((x + y)^{n + 2}) = (n + 2)(x + y)^{n + 1} \left(1 + \frac{dy}{dx}\right) \]

3. **Substituting \(\frac{dy}{dx}\)**:
Substitute \(\frac{dy}{dx} = \frac{y}{x}\) into the differentiated equations:

- **Left-hand Side**:
\[ 2xy^3 + 3x^2y^2 \left(\frac{y}{x}\right) = 2xy^3 + 3xy^2y = 2xy^3 + 3xy^3 = 5xy^3 \]

- **Right-hand Side**:
\[ (n + 2)(x + y)^{n + 1} \left(1 + \frac{y}{x}\right) = (n + 2)(x + y)^{n + 1} \left(\frac{x + y}{x}\right) = (n + 2)(x + y)^{n + 1} \frac{x + y}{x} \]

4. **Equating Both Sides**:
Now we equate the left-hand side and right-hand side:
\[ 5xy^3 = (n + 2)(x + y)^{n + 1} \frac{x + y}{x} \]

5. **Simplifying the Equation**:
Rearranging gives:
\[ 5xy^3 = (n + 2)(x + y)^{n + 2} \frac{1}{x} \]
Multiplying through by \(x\):
\[ 5x^2y^3 = (n + 2)(x + y)^{n + 2} \]

6. **Comparing Coefficients**:
Since we know \(x^2 y^3\) is on both sides, we can compare coefficients:
\[ 5 = n + 2 \]

7. **Solving for \(n\)**:
\[ n = 5 - 2 = 3 \]

Thus, the value of \(n\) is:
\[ \boxed{3} \]