Which of the following differential equations has $y = {c_1}{e^x} + {c_2}{e^{ - x}}$ as the general solution?

• $\cfrac{{{d^2}y}}{{d{x^2}}} + y = 0$

• $\cfrac{{{d^2}y}}{{d{x^2}}} - y = 0$

• $\cfrac{{{d^2}y}}{{d{x^2}}} + 1 = 0$

• $\cfrac{{{d^2}y}}{{d{x^2}}} - 1 = 0$

.: (B) Given general solution is $y = {c_1}{e^x} + {c_2}{e^{ - x}}$ …(1)

Differentiating (1) w.r.t. $x$,

we get
${y_1} = {c_1}{e^x} + {c_2}{e^{ - x}}( - 1)$
…(2)
Again differentiating (2) w.r.t. $x$,

we get
${y_2} = {c_1}{e^x} - {c_2}{e^{ - x}}( - 1) \Rightarrow {y_2} = {c_1}{e^x} + {c_2}{e^{ - x}}$

$\Rightarrow {y_2} = y$

(using (1))