Question

Which of the following differential equations has y = C₁ e^x + C₂ e^−x as the general solution?

Options

• \[\frac{d^2 y}{d x^2} + y = 0\]

• \[\frac{d^2 y}{d x^2} - y = 0\]

• \[\frac{d^2 y}{d x^2} + 1 = 0\]

• \[\frac{d^2 y}{d x^2} - 1 = 0\]

Answer 1

\[\frac{d^2 y}{d x^2} - y = 0\]

\[y = C_1 e^x + C_2 e^{- x} . . . . . \left( 1\right)\]

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

\[\frac{d^2 y}{d x^2} = C_1 e^x + C_2 e^{- x} \]

\[ \Rightarrow \frac{d^2 y}{d x^2} = y ...........\left[\text{Using }\left( 1 \right)\text{ and }\left( 2 \right) \right]\]

\[ \Rightarrow \frac{d^2 y}{d x^2} - y = 0\]