The straight line `y=2x` meets `y=f(x)` at P, where f(x) is a solution of the differential equation `(dy)/(dx)=(x^(2)+xy)/(x^(2)+y^(2))` such that `f(1)=3`, then `f'(x)` at point P is

Options

A. `(3)/(5)`
B. `(5)/(3)`
C. 2
D. 1

Answer

To solve the problem, we need to find \( f'(x) \) at the point \( P \) where the line \( y = 2x \) intersects the curve \( y = f(x) \), given that \( f(x) \) satisfies the differential equation:

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

and that \( f(1) = 3 \).

### Step 1: Substitute \( y = 2x \) into the differential equation

We start by substituting \( y = 2x \) into the differential equation to find the slope at point \( P \).

\[ \frac{dy}{dx} = \frac{x^2 + 2x^2}{x^2 + (2x)^2} = \frac{3x^2}{x^2 + 4x^2} = \frac{3x^2}{5x^2} = \frac{3}{5} \]

### Step 2: Find the coordinates of point \( P \)

Next, we need to find the coordinates of point \( P \) where \( y = f(x) \) intersects \( y = 2x \). Since we know \( f(1) = 3 \), we can check if \( (1, 3) \) is on the line \( y = 2x \).

Calculating \( y \) when \( x = 1 \):

\[ y = 2(1) = 2 \]

Since \( f(1) = 3 \) does not equal \( 2 \), we need to find another point of intersection.

### Step 3: Set \( f(x) = 2x \) and solve for \( x \)

To find the intersection, we need to set \( f(x) = 2x \) and solve for \( x \):

1. Substitute \( y = f(x) \) into the differential equation.
2. We already know that \( f(x) \) must intersect the line \( y = 2x \) at some point, so we need to find that point.

### Step 4: Find \( f'(x) \) at point \( P \)

From our previous calculation, we found that at point \( P \) where \( y = 2x \):

\[ f'(x) = \frac{dy}{dx} = \frac{3}{5} \]

Thus, the value of \( f'(x) \) at point \( P \) is:

\[ \boxed{\frac{3}{5}} \]