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Which of the following differential equa...

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

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`

Text Solution

AI Generated Solution

The correct Answer is:
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**.
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