A solution of the differential equation, `((dy) /( dx))^2- x ( dy ) /( dx ) + y=0`

Options

A. `y=2`
B. `4y =x^2`
C. `y= 2x -4`
D. `y= 2x ^2 -4`

Answer

To solve the differential equation \(\left(\frac{dy}{dx}\right)^2 - x \frac{dy}{dx} + y = 0\), we will follow these steps:

### Step 1: Substitute \( \frac{dy}{dx} \) with \( p \)
Let \( p = \frac{dy}{dx} \). Then, we can rewrite the equation as:
\[ p^2 - xp + y = 0 \]

### Step 2: Solve for \( y \)
Rearranging the equation gives us:
\[ y = xp - p^2 \]

### Step 3: Differentiate \( y \) with respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(xp - p^2) \]

Using the product rule and chain rule, we get:
\[ \frac{dy}{dx} = \frac{d}{dx}(xp) - \frac{d}{dx}(p^2) \]
\[ = p + x\frac{dp}{dx} - 2p\frac{dp}{dx} \]
\[ = p + (x - 2p)\frac{dp}{dx} \]

### Step 4: Set the two expressions for \( \frac{dy}{dx} \) equal
Since we have assumed \( \frac{dy}{dx} = p \), we can set the two expressions equal:
\[ p = p + (x - 2p)\frac{dp}{dx} \]

### Step 5: Simplify the equation
Subtract \( p \) from both sides:
\[ 0 = (x - 2p)\frac{dp}{dx} \]

### Step 6: Analyze the equation
This equation implies either \( \frac{dp}{dx} = 0 \) or \( x - 2p = 0 \).

1. **Case 1:** If \( \frac{dp}{dx} = 0 \), then \( p \) is a constant. Let \( p = c \), where \( c \) is a constant.

2. **Case 2:** If \( x - 2p = 0 \), then \( p = \frac{x}{2} \).

### Step 7: Find \( y \) for Case 1
Substituting \( p = c \) into the expression for \( y \):
\[ y = xc - c^2 \]

### Step 8: Find \( y \) for Case 2
Substituting \( p = \frac{x}{2} \) into the expression for \( y \):
\[ y = x\left(\frac{x}{2}\right) - \left(\frac{x}{2}\right)^2 \]
\[ = \frac{x^2}{2} - \frac{x^2}{4} = \frac{x^2}{4} \]

### Conclusion
The general solution from Case 1 is:
\[ y = xc - c^2 \]
This can be expressed as:
\[ y = cx - c^2 \]
where \( c \) is a constant.

### Final Result
Thus, the solution of the given differential equation is:
\[ y = xc - c^2 \]