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
We start with the given equation:
\[ x \frac{dy}{dx} + 2y = x^2 \]
We can divide the entire equation by \( x \) (since \( x \neq 0 \)):
\[ \frac{dy}{dx} + \frac{2}{x} y = x \]

### Step 2: Identify \( p \) and \( q \)
Now, we can identify \( p \) and \( q \) from the standard form \( \frac{dy}{dx} + p y = q \):
- \( p = \frac{2}{x} \)
- \( q = x \)

### 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 through 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:
\[ \frac{d}{dx}(y \cdot x^2) = x^3 \]

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

### Step 7: Solve for \( y \)
Now, we can 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 use the initial condition \( y(1) = 1 \):
\[ 1 = \frac{1^2}{4} + \frac{C}{1^2} \]
This simplifies to:
\[ 1 = \frac{1}{4} + C \implies C = 1 - \frac{1}{4} = \frac{3}{4} \]

### Step 9: Substitute \( C \) back into the equation
Now substituting \( C \) back into the equation for \( y \):
\[ y = \frac{x^2}{4} + \frac{3/4}{x^2} \]

### Final Solution
Thus, the final solution to the differential equation is:
\[ y = \frac{x^2}{4} + \frac{3}{4x^2} \]