If `y=f (x)` satisfy the differential equation `(dy)/(dx) + y/x =x ^(2),f (1)=1,` then value of `f (3)` equals:

Options

A. 7
B. 5
C. 9
D. 27

Answer

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

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

### Step 2: Find the integrating factor
The integrating factor \(\mu(x)\) is given by:
\[ \mu(x) = e^{\int P(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x| \]
Since \(x\) is positive in our context, we can simplify this to:
\[ \mu(x) = x \]

### Step 3: Multiply the differential equation by the integrating factor
Multiplying the entire differential equation by the integrating factor \(x\):
\[ x \frac{dy}{dx} + y = x^3 \]

### Step 4: Rewrite the left side as a derivative
The left side can be rewritten as:
\[ \frac{d}{dx}(xy) = x^3 \]

### Step 5: Integrate both sides
Integrating both sides with respect to \(x\):
\[ \int \frac{d}{dx}(xy) \, dx = \int x^3 \, dx \]
This gives us:
\[ xy = \frac{x^4}{4} + C \]
where \(C\) is the constant of integration.

### Step 6: Solve for \(y\)
Now, we can express \(y\) (which is \(f(x)\)):
\[ y = \frac{x^4}{4x} + \frac{C}{x} = \frac{x^3}{4} + \frac{C}{x} \]

### Step 7: Use the initial condition to find \(C\)
Using the initial condition \(f(1) = 1\):
\[ 1 = \frac{1^3}{4} + \frac{C}{1} \]
This simplifies to:
\[ 1 = \frac{1}{4} + C \implies C = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 8: Substitute \(C\) back into the equation for \(y\)
Substituting \(C\) back, we have:
\[ f(x) = \frac{x^3}{4} + \frac{3/4}{x} = \frac{x^3}{4} + \frac{3}{4x} \]

### Step 9: Find \(f(3)\)
Now we need to find \(f(3)\):
\[ f(3) = \frac{3^3}{4} + \frac{3}{4 \cdot 3} = \frac{27}{4} + \frac{3}{12} \]
Converting \(\frac{3}{12}\) to have a common denominator:
\[ \frac{3}{12} = \frac{1}{4} \]
Thus,
\[ f(3) = \frac{27}{4} + \frac{1}{4} = \frac{28}{4} = 7 \]

### Final Answer
Therefore, the value of \(f(3)\) is:
\[ \boxed{7} \]