Question:
If \( y_1 \) and \( y_2 \) are two different solutions of the ordinary differential equation
\[
y'' + \sin(e^x)y = \cos(e^x), \quad 0<x<1,
\]
then which one of the following is its general solution on \( [0, 1] \)?
If \( y_1 \) and \( y_2 \) are two different solutions of the ordinary differential equation
\[
y'' + \sin(e^x)y = \cos(e^x), \quad 0<x<1,
\]
then which one of the following is its general solution on \( [0, 1] \)?
Show Hint
For second-order ODEs, verify that the proposed solution includes arbitrary constants and satisfies the linearity of the equation.
Updated On: Feb 1, 2025
- \( c_1 y_1 + c_2 y_2, \, c_1, c_2 \in {R} \)
- \( y_1 + c(e^{x} - y_2), \, c \in {R} \)
- \( e^x y_1 + c(e^{-x} - y_2), \, c \in {R} \)
- \( c_1(y_1 + y_2) + c_2(y_1 - y_2), \, c_1, c_2 \in {R} \)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Structure of the solution.
For a second-order linear differential equation, the general solution is a linear combination of independent solutions.
Step 2: Analyzing the given options.
- \( y_1 \) and \( y_2 \) are independent solutions. The correct form for a general solution incorporates these and an arbitrary constant \( c \).
- Option (2) satisfies this structure as \( y_1 + c(e^x - y_2) \).
Step 3: Conclusion.
The correct answer is \( {(2)} \).
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