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

Options

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

Answer

To solve the differential equation

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

we can follow these steps:

### Step 1: Introduce a substitution
Let \( p = \frac{dy}{dx} \). Then, we can rewrite the equation as:

\[ p^2 - xp + y = 0. \]

### Step 2: Solve for \( y \)
This is a quadratic equation in terms of \( p \). We can use the quadratic formula to solve for \( p \):

\[ p = \frac{x \pm \sqrt{x^2 - 4y}}{2}. \]

### Step 3: Express \( y \) in terms of \( p \)
Rearranging the equation gives us:

\[ y = \frac{p^2 - xp}{1}. \]

### Step 4: Differentiate \( y \) with respect to \( x \)
Now we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = p = \frac{d}{dx}\left(\frac{p^2 - xp}{1}\right). \]

### Step 5: Substitute back the expression for \( p \)
From our earlier expression, we can substitute \( p \) back into the equation. This will yield:

\[ p = \frac{x \pm \sqrt{x^2 - 4y}}{2}. \]

### Step 6: Find the general solution
To find the general solution, we can express \( y \) in terms of \( p \):

\[ y = xp - p^2. \]

### Step 7: Identify the constant
Since \( p \) is a constant, we can write:

\[ y = xc - c^2, \]

where \( c \) is a constant.

### Step 8: Final form of the solution
Thus, the solution of the differential equation is:

\[ y = xc - c^2, \]

where \( c \) is an arbitrary constant.