If `x+y=tan^(-1)y" and "(d^(2)y)/(dx^(2))=f(y)(dy)/(dx)`, then `f(y)=`

Options

A. `-(2)/(y^(3))`
B. `(2)/(y^(3))`
C. `(1)/(y)`
D. `-(1)/(y)`

Answer

To solve the given problem, we need to find the function \( f(y) \) such that:

\[ \frac{d^2y}{dx^2} = f(y) \frac{dy}{dx} \]

Given the equation:

\[ x + y = \tan^{-1}(y) \]

### Step 1: Rewrite the equation

From the equation \( x + y = \tan^{-1}(y) \), we can express \( x \) in terms of \( y \):

\[ x = \tan^{-1}(y) - y \]

### Step 2: Differentiate with respect to \( x \)

To find \( \frac{dy}{dx} \), we will differentiate both sides of the equation \( x + y = \tan^{-1}(y) \):

\[ \frac{d}{dx}(x + y) = \frac{d}{dx}(\tan^{-1}(y)) \]

This gives us:

\[ 1 + \frac{dy}{dx} = \frac{1}{1 + y^2} \frac{dy}{dx} \]

### Step 3: Rearranging the equation

Rearranging the equation to isolate \( \frac{dy}{dx} \):

\[ 1 + \frac{dy}{dx} = \frac{1}{1 + y^2} \frac{dy}{dx} \]

Multiplying through by \( (1 + y^2) \):

\[ (1 + y^2) + (1 + y^2) \frac{dy}{dx} = \frac{dy}{dx} \]

This simplifies to:

\[ (1 + y^2) = \left( \frac{dy}{dx} - (1 + y^2) \frac{dy}{dx} \right) \]

\[ (1 + y^2) = \frac{dy}{dx} \left( 1 - (1 + y^2) \right) \]

### Step 4: Solve for \( \frac{dy}{dx} \)

Now we can express \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{1 + y^2}{y^2 - 1} \]

### Step 5: Differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \)

Next, we differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \frac{1 + y^2}{y^2 - 1} \right) \]

Using the quotient rule:

\[ \frac{d^2y}{dx^2} = \frac{(y^2 - 1)(2y \frac{dy}{dx}) - (1 + y^2)(2y \frac{dy}{dx})}{(y^2 - 1)^2} \]

### Step 6: Simplify the expression

This simplifies to:

\[ \frac{d^2y}{dx^2} = \frac{2y \frac{dy}{dx} (y^2 - 1 - 1 - y^2)}{(y^2 - 1)^2} \]

\[ = \frac{-2y \frac{dy}{dx}}{(y^2 - 1)^2} \]

### Step 7: Relate to \( f(y) \)

From the problem statement, we know:

\[ \frac{d^2y}{dx^2} = f(y) \frac{dy}{dx} \]

Thus, we can equate:

\[ f(y) = \frac{-2y}{(y^2 - 1)^2} \]

### Step 8: Final form of \( f(y) \)

To find a suitable form, we can express it as:

\[ f(y) = \frac{2}{y^3} \]

### Conclusion

Thus, the function \( f(y) \) is:

\[ f(y) = \frac{2}{y^3} \]