Let y=f(x) be a real valued function satisfying `xdy/dx = x^2 + y-2`, f(1)=1 then f(3) equal

Options

A. 8
B. 7
C. 5
D. 2

Answer

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

### Step 1: Rewrite the Differential Equation
We start with the equation:
\[ x \frac{dy}{dx} = x^2 + y - 2 \]
We can rearrange it to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{x^2 + y - 2}{x} \]
This simplifies to:
\[ \frac{dy}{dx} = x + \frac{y}{x} - \frac{2}{x} \]

### Step 2: Identify the Form of the Equation
The equation can be rewritten as:
\[ \frac{dy}{dx} - \frac{y}{x} = x - \frac{2}{x} \]
This is a linear first-order differential equation of the form:
\[ \frac{dy}{dx} + P(x)y = Q(x) \]
where \( P(x) = -\frac{1}{x} \) and \( Q(x) = x - \frac{2}{x} \).

### Step 3: Find the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int -\frac{1}{x} \, dx} = e^{-\ln|x|} = \frac{1}{x} \]

### Step 4: Multiply the Equation by the Integrating Factor
Multiplying the entire differential equation by the integrating factor \( \frac{1}{x} \):
\[ \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = 1 - \frac{2}{x^2} \]

### Step 5: Rewrite the Left Side
The left side can be expressed as:
\[ \frac{d}{dx} \left( \frac{y}{x} \right) = 1 - \frac{2}{x^2} \]

### Step 6: Integrate Both Sides
Integrating both sides gives:
\[ \frac{y}{x} = \int \left( 1 - \frac{2}{x^2} \right) dx \]
Calculating the integral:
\[ \int \left( 1 - \frac{2}{x^2} \right) dx = x + \frac{2}{x} + C \]
Thus, we have:
\[ \frac{y}{x} = x + \frac{2}{x} + C \]

### Step 7: Solve for \( y \)
Multiplying through by \( x \):
\[ y = x^2 + 2 + Cx \]

### Step 8: Apply the Initial Condition
Given \( f(1) = 1 \):
\[ 1 = 1^2 + 2 + C(1) \]
This simplifies to:
\[ 1 = 1 + 2 + C \implies C = -2 \]

### Step 9: Substitute \( C \) Back into the Equation
Substituting \( C \) back:
\[ y = x^2 + 2 - 2x \]
Thus, we have:
\[ f(x) = x^2 - 2x + 2 \]

### Step 10: Find \( f(3) \)
Now we calculate \( f(3) \):
\[ f(3) = 3^2 - 2(3) + 2 = 9 - 6 + 2 = 5 \]

### Final Answer
Thus, the value of \( f(3) \) is:
\[ \boxed{5} \]