Question

If y\(_1\) and y\(_2\) be two solutions of the differential equation \(\frac{d^2y}{dx^2} + 2y = f(x)\), then y\(_1\) + y\(_2\) is also solution of

• A. \(\frac{d^2y}{dx^2} + 2y = 0\)
• B. \(\frac{d^2y}{dx^2} + 2y = 2f(x)\)
• C. \(\frac{d^2y}{dx^2} + 2y = 4f(x)\)
• D. \(\frac{d^2y}{dx^2} + 2y = 5f(x)\)

Hint:

Linear Operator Rules:
- If \( L(y) = f(x) \), then the sum of two solutions \( y_1 + y_2 \) satisfies \( L(y_1 + y_2) = 2f(x) \).
- The difference \( y_1 - y_2 \) satisfies the homogeneous equation \( L(y_1 - y_2) = 0 \).
This basic algebraic property allows you to solve these problems without doing any calculus.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This question tests the principle of superposition and linearity of ordinary differential equations.
We are given a second-order linear non-homogeneous differential equation and two specific solutions, and we need to find the equation satisfied by their sum.

Step 2: Key Formula or Approach:
Let \( L \) be a linear differential operator. For our equation:
\[ L(y) = \frac{d^2y}{dx^2} + 2y \]
The linearity property of the operator states that:
\[ L(c_1 y_1 + c_2 y_2) = c_1 L(y_1) + c_2 L(y_2) \]
for any constants \( c_1, c_2 \) and functions \( y_1, y_2 \).

Step 3: Detailed Explanation:

• Given Solutions:
- Since \( y_1 \) is a solution of \( L(y) = f(x) \):
\[ L(y_1) = \frac{d^2y_1}{dx^2} + 2y_1 = f(x) \]
- Since \( y_2 \) is also a solution of \( L(y) = f(x) \):
\[ L(y_2) = \frac{d^2y_2}{dx^2} + 2y_2 = f(x) \]

• Evaluate the Operator for \( y_1 + y_2 \):
- Using the linearity of the operator \( L \):
\[ L(y_1 + y_2) = L(y_1) + L(y_2) \]
- Substitute the known values of \( L(y_1) \) and \( L(y_2) \):
\[ L(y_1 + y_2) = f(x) + f(x) = 2f(x) \]
- Write out the expanded form:
\[ \frac{d^2(y_1 + y_2)}{dx^2} + 2(y_1 + y_2) = 2f(x) \]
- This shows that \( y_1 + y_2 \) is a solution of the equation \( \frac{d^2y}{dx^2} + 2y = 2f(x) \).


Step 4: Final Answer:
The sum \( y_1 + y_2 \) satisfies the differential equation \( \frac{d^2y}{dx^2} + 2y = 2f(x) \).
Therefore, the correct choice is option (B).