The solution of the differential equation ` x(dy)/(dx) + 2y = x^2 (X ne 0)` with ` y(1) =1,` is :

Options

A. `y= (4)/(5)x^3+(1)/(5x^2)`
B. `y= (x^3)/(5)+ (1)/(5x ^2)`
C. ` y=( x^2 )/(4) +(3)/(4x ^2)`
D. `y=(3)/(4)x^2 +(1)/(4x^2)`

Answer

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

### Step 1: Rewrite the Equation
First, we rewrite the given differential equation in a more standard form. We divide the entire equation by \( x \) (since \( x \neq 0 \)):
\[ \frac{dy}{dx} + \frac{2y}{x} = x \]

### Step 2: Identify \( p \) and \( q \)
From the rewritten equation, we can identify:
- \( p = \frac{2}{x} \)
- \( q = x \)

### Step 3: Find the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p \, dx} = e^{\int \frac{2}{x} \, dx} \]
Calculating the integral:
\[ \int \frac{2}{x} \, dx = 2 \ln |x| = \ln |x|^2 \]
Thus, the integrating factor becomes:
\[ \mu(x) = e^{\ln |x|^2} = x^2 \]

### Step 4: Multiply the Equation by the Integrating Factor
Now, we multiply the entire differential equation by the integrating factor \( x^2 \):
\[ x^2 \frac{dy}{dx} + 2xy = x^3 \]

### Step 5: Rewrite the Left Side as a Derivative
The left side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(x^2 y) = x^3 \]

### Step 6: Integrate Both Sides
Next, we integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(x^2 y) \, dx = \int x^3 \, dx \]
This gives us:
\[ x^2 y = \frac{x^4}{4} + C \]

### Step 7: Solve for \( y \)
Now, we solve for \( y \):
\[ y = \frac{x^4}{4x^2} + \frac{C}{x^2} = \frac{x^2}{4} + \frac{C}{x^2} \]

### Step 8: Apply the Initial Condition
We apply the initial condition \( y(1) = 1 \):
\[ 1 = \frac{1^2}{4} + \frac{C}{1^2} \]
This simplifies to:
\[ 1 = \frac{1}{4} + C \]
Thus, solving for \( C \):
\[ C = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 9: Write the Final Solution
Substituting \( C \) back into the equation for \( y \):
\[ y = \frac{x^2}{4} + \frac{3/4}{x^2} = \frac{x^2}{4} + \frac{3}{4x^2} \]

### Final Answer
The solution of the differential equation is:
\[ y = \frac{x^2}{4} + \frac{3}{4x^2} \]

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