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
- \( c_1 y_1 + c_2 y_2, \, c_1, c_2 \in {R} \)
- \( y_1 + c(e^{x} - y_2), \, c \in {R} \)
- \( e^x y_1 + c(e^{-x} - y_2), \, c \in {R} \)
- \( c_1(y_1 + y_2) + c_2(y_1 - y_2), \, c_1, c_2 \in {R} \)
The Correct Option is B
Solution and Explanation
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