If `y = y(x)` is the solution of the differential equation,`x (dy)/(dx) + 5y = 10x^5, x > 0` satisfying `y(1) = 2` then range of `y(x)` for `x > 0` is:

Options

A. `[1,oo)`
B. `[2 , oo)`
C. `(2, oo)`
D. `[4,00)`

Explanation

To solve the differential equation \( x \frac{dy}{dx} + 5y = 10x^5 \) with the initial condition \( y(1) = 2 \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given equation:
\[ x \frac{dy}{dx} + 5y = 10x^5 \]
We can rewrite it in standard linear form:
\[ \frac{dy}{dx} + \frac{5}{x} y = 10x^4 \]

### Step 2: Identify the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int \frac{5}{x} dx} = e^{5 \ln x} = x^5 \]

### Step 3: Multiply the Equation by the Integrating Factor
Now we multiply the entire equation by the integrating factor \( x^5 \):
\[ x^5 \frac{dy}{dx} + 5x^4 y = 10x^9 \]
This can be rewritten as:
\[ \frac{d}{dx}(x^5 y) = 10x^9 \]

### Step 4: Integrate Both Sides
Integrating both sides with respect to \( x \):
\[ \int \frac{d}{dx}(x^5 y) \, dx = \int 10x^9 \, dx \]
This gives us:
\[ x^5 y = x^{10} + C \]
where \( C \) is the constant of integration.

### Step 5: Solve for \( y \)
Now, we solve for \( y \):
\[ y = \frac{x^{10} + C}{x^5} = x^5 + \frac{C}{x^5} \]

### Step 6: Apply the Initial Condition
We use the initial condition \( y(1) = 2 \):
\[ 2 = 1^5 + \frac{C}{1^5} \implies 2 = 1 + C \implies C = 1 \]

### Step 7: Write the Final Solution
Thus, the solution to the differential equation is:
\[ y = x^5 + \frac{1}{x^5} \]

### Step 8: Find the Range of \( y(x) \)
To find the range of \( y(x) = x^5 + \frac{1}{x^5} \) for \( x > 0 \), we can use the AM-GM inequality:
\[ \frac{x^5 + \frac{1}{x^5}}{2} \geq \sqrt{x^5 \cdot \frac{1}{x^5}} = 1 \]
Thus,
\[ x^5 + \frac{1}{x^5} \geq 2 \]
The minimum value occurs when \( x^5 = \frac{1}{x^5} \) or \( x = 1 \), giving \( y(1) = 2 \).

As \( x \to 0^+ \), \( y \to \infty \) and as \( x \to \infty \), \( y \to \infty \).

### Conclusion
The range of \( y(x) \) for \( x > 0 \) is:
\[ [2, \infty) \]