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Question
Which of the following differential equations has y = C1 ex + C2 e−x as the general solution?
Options
\[\frac{d^2 y}{d x^2} + y = 0\]
\[\frac{d^2 y}{d x^2} - y = 0\]
\[\frac{d^2 y}{d x^2} + 1 = 0\]
\[\frac{d^2 y}{d x^2} - 1 = 0\]
Solution
\[\frac{d^2 y}{d x^2} - y = 0\]
\[y = C_1 e^x + C_2 e^{- x} . . . . . \left( 1\right)\]
Differentiating both sides of (1) with respect to x, we get
\[\frac{dy}{dx} = C_1 e^x - C_2 e^{- x} . . . . . \left( 2 \right)\]
Differentiating both sides of (2) with respect to x, we get
\[\frac{d^2 y}{d x^2} = C_1 e^x + C_2 e^{- x} \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = y ...........\left[\text{Using }\left( 1 \right)\text{ and }\left( 2 \right) \right]\]
\[ \Rightarrow \frac{d^2 y}{d x^2} - y = 0\]
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