The solution of `((dy)/(dx))^(2)-2(x+(1)/(4x))(dy)/(dx)+1=0`

Options

A. `y = x^(2) + c`
B. `y = (1)/(2) In (x) + c, x gt 0`
C. `y = (x)/(2) + c`
D. `y = (x^(2))/(2) + c`

Explanation

To solve the differential equation

\[ \left(\frac{dy}{dx}\right)^2 - 2\left(x + \frac{1}{4x}\right)\frac{dy}{dx} + 1 = 0, \]

we can start by treating it as a quadratic equation in terms of \(\frac{dy}{dx}\).

### Step 1: Identify the quadratic form

The equation can be rewritten as:

\[ a\left(\frac{dy}{dx}\right)^2 + b\left(\frac{dy}{dx}\right) + c = 0, \]

where:
- \(a = 1\),
- \(b = -2\left(x + \frac{1}{4x}\right)\),
- \(c = 1\).

### Step 2: Factor the quadratic equation

To factor the quadratic equation, we can use the quadratic formula:

\[ \frac{dy}{dx} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]

Calculating \(b^2 - 4ac\):

\[ b^2 = \left(-2\left(x + \frac{1}{4x}\right)\right)^2 = 4\left(x + \frac{1}{4x}\right)^2, \]
\[ 4ac = 4 \cdot 1 \cdot 1 = 4. \]

Thus,

\[ b^2 - 4ac = 4\left(x + \frac{1}{4x}\right)^2 - 4. \]

### Step 3: Solve for \(\frac{dy}{dx}\)

Now we can find the roots:

\[ \frac{dy}{dx} = \frac{2\left(x + \frac{1}{4x}\right) \pm 2\sqrt{\left(x + \frac{1}{4x}\right)^2 - 1}}{2}. \]

This simplifies to:

\[ \frac{dy}{dx} = \left(x + \frac{1}{4x}\right) \pm \sqrt{\left(x + \frac{1}{4x}\right)^2 - 1}. \]

### Step 4: Find the two solutions

We can express the two solutions as:

1. \(\frac{dy}{dx} = 2x\)
2. \(\frac{dy}{dx} = \frac{1}{2x}\)

### Step 5: Integrate both equations

**For the first equation:**

\[ \frac{dy}{dx} = 2x \implies dy = 2x \, dx. \]

Integrating both sides:

\[ y = x^2 + C_1. \]

**For the second equation:**

\[ \frac{dy}{dx} = \frac{1}{2x} \implies dy = \frac{1}{2x} \, dx. \]

Integrating both sides:

\[ y = \frac{1}{2} \ln |x| + C_2. \]

### Final Solutions

Thus, the solutions to the differential equation are:

1. \(y = x^2 + C_1\)
2. \(y = \frac{1}{2} \ln |x| + C_2\) (where \(x > 0\)).