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 will follow these steps:

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

### Step 2: Rearrange the equation to express \(y\) in terms of \(p\)
From the equation, we can express \(y\) as:
\[ y = xp - p^2 \]

### 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 on \(xp\):
\[ \frac{dy}{dx} = p + x\frac{dp}{dx} - 2p\frac{dp}{dx} \]
This simplifies to:
\[ \frac{dy}{dx} = p + (x - 2p)\frac{dp}{dx} \]

### Step 4: Set \(\frac{dy}{dx}\) equal to \(p\)
Since we defined \(p = \frac{dy}{dx}\), we can set:
\[ p + (x - 2p)\frac{dp}{dx} = p \]
Subtracting \(p\) from both sides gives:
\[ (x - 2p)\frac{dp}{dx} = 0 \]

### Step 5: Analyze the equation
The equation \((x - 2p)\frac{dp}{dx} = 0\) implies either:
1. \(x - 2p = 0\) or
2. \(\frac{dp}{dx} = 0\)

If \(\frac{dp}{dx} = 0\), then \(p\) is a constant.

### Step 6: Solve for \(y\) when \(p\) is constant
If \(p\) is constant, we can denote it as \(c\):
\[ y = cx - c^2 \]

### Step 7: Identify the solution
This indicates that the general solution of the differential equation is:
\[ y = cx - c^2 \]
where \(c\) is a constant.

### Step 8: Check the options
Now, we need to check which of the given options corresponds to the form \(y = cx - c^2\).

### Conclusion
Upon checking the options provided in the question, we find that the third option \(y = 2x - 4\) corresponds to this form when \(c = 2\) (since \(c^2 = 4\)).

Thus, the correct answer is:
\[ \text{Option 3: } y = 2x - 4 \]

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