Solve the equation `(dy)/(dx)=1-1/x^(2)`..... if f(2)=7..............

Explanation

To solve the differential equation \(\frac{dy}{dx} = 1 - \frac{1}{x^2}\) with the initial condition \(f(2) = 7\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \frac{dy}{dx} = 1 - \frac{1}{x^2} \]

### Step 2: Separate variables
We can rewrite this as:
\[ dy = \left(1 - \frac{1}{x^2}\right)dx \]

### Step 3: Integrate both sides
Now we integrate both sides:
\[ \int dy = \int \left(1 - \frac{1}{x^2}\right)dx \]

The left side integrates to:
\[ y \]

The right side integrates as follows:
\[ \int 1 \, dx - \int \frac{1}{x^2} \, dx = x + \frac{1}{x} + C \]
where \(C\) is the constant of integration.

Thus, we have:
\[ y = x + \frac{1}{x} + C \]

### Step 4: Apply the initial condition
We know that \(f(2) = 7\), which means:
\[ y(2) = 7 \]

Substituting \(x = 2\) into the equation:
\[ 7 = 2 + \frac{1}{2} + C \]

Calculating the right side:
\[ 7 = 2 + 0.5 + C \implies 7 = 2.5 + C \implies C = 7 - 2.5 = 4.5 \]

### Step 5: Write the final solution
Now substituting \(C\) back into the equation gives us:
\[ y = x + \frac{1}{x} + 4.5 \]

### Final Answer
Thus, the solution to the differential equation is:
\[ y = x + \frac{1}{x} + 4.5 \]