Which of the following differential equations has y = x as one of its particular solution?

Options

A. `(d^2y)/(dx^2)-x^2(dy)/(dx)+xy=x`
B. `(d^2y)/(dx^2)+x(dy)/(dx)+xy=x`
C. `(d^2y)/(dx^2)-x^2(dy)/(dx)+xy=0`
D. `(d^2y)/(dx^2)+x(dy)/(dx)+xy=0`

Answer

To determine which of the given differential equations has \( y = x \) as a particular solution, we will analyze each option step by step.

### Step 1: Identify the Particular Solution
We are given that \( y = x \) is a particular solution. This means that for this solution:
- \( \frac{dy}{dx} = 1 \)
- \( \frac{d^2y}{dx^2} = 0 \)

### Step 2: Evaluate Each Option

#### Option 1:
\[ \frac{d^2y}{dx^2} - x^2 + y^2 = 0 \]
Substituting \( y = x \):
- \( \frac{d^2y}{dx^2} = 0 \)
- \( x^2 = x^2 \)
- \( y^2 = x^2 \)

Substituting these into the equation:
\[ 0 - x^2 + x^2 = 0 \]
This simplifies to \( 0 = 0 \), which is true. Thus, **Option 1 is a valid solution**.

#### Option 2:
\[ \frac{d^2y}{dx^2} - x + y = 0 \]
Substituting \( y = x \):
- \( \frac{d^2y}{dx^2} = 0 \)
- \( -x + x = 0 \)

Substituting these into the equation:
\[ 0 - x + x = 0 \]
This simplifies to \( 0 = 0 \), which is true. Thus, **Option 2 is also a valid solution**.

#### Option 3:
\[ \frac{d^2y}{dx^2} - x^2 + \left(\frac{dy}{dx}\right)^2 + xy = 0 \]
Substituting \( y = x \):
- \( \frac{d^2y}{dx^2} = 0 \)
- \( \frac{dy}{dx} = 1 \)

Substituting these into the equation:
\[ 0 - x^2 + (1)^2 + x \cdot x = 0 \]
This simplifies to:
\[ 0 - x^2 + 1 + x^2 = 0 \]
This simplifies to \( 1 = 0 \), which is false. Thus, **Option 3 is not a valid solution**.

### Conclusion
The differential equations that have \( y = x \) as a particular solution are **Option 1 and Option 2**.