Which of the following differential equation has $y = c_{1} e^{x} + c_{2} e^{- x}$ as the general solution?

\frac{d^{2} y}{d x^{2}} + y = 0

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

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\frac{d^{2} y}{d x^{2}} + 1 = 0

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\frac{d^{2} y}{d x^{2}} - 1 = 0

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Solution

The correct option is B $\frac{d^{2} y}{d x^{2}} - y = 0$
$\frac{d y}{d x} = c_{1} e^{x} - c_{2} e^{- x} \frac{d^{2} y}{d x^{2}} = c_{1} e^{x} - c_{2} e^{- x} = y \frac{d^{2} y}{d x^{2}} - y = 0$

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Similar questions

Which of the following differential equations has $y = c_{1} e^{x} + c_{2} e^{- x}$ as the general solution?
(a) $\frac{d^{2} y}{d x^{2}} + y = 0$
(b) $\frac{d^{2} y}{d x^{2}} - y = 0$
(c) $\frac{d^{2} y}{d x^{2}} + 1 = 0$
(d) $\frac{d^{2} y}{d x^{2}} - 1 = 0$

Q. Show that

y = c_{1} e^{x} + c_{2} e^{- x}

is the general solution of the differential equation

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

The differential equation for the equation y = Acos αx + Bsin αx is:

Which of the following differential equations has y=x as one of its particular solution?
(a) $\frac{d^{2} y}{d x^{2}} - x^{2} \frac{d y}{d x} + x y = x$
(b) $\frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + x y = x$
(c) $\frac{d^{2} y}{d x^{2}} - x^{2} \frac{d y}{d x} + x y = 0$
(d) $\frac{d^{2} y}{d x^{2}} + x \frac{d y}{d x} + x y = 0$

The differential equation whose solution is $y = c_{1} c o s a x + c_{2} s i n a x$ is
(Where $c_{1}, c_{2}$ are arbitrary constants)

[MP PET 1996]

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Which of the following differential equation has y=c1ex+c2e−x as the general solution?

The correct option is B d2ydx2−y=0dydx=c1ex−c2e−xd2ydx2=c1ex−c2e−x=yd2ydx2−y=0

Which of the following differential equation has $y = c_{1} e^{x} + c_{2} e^{- x}$ as the general solution?

The correct option is B $\frac{d^{2} y}{d x^{2}} - y = 0$
$\frac{d y}{d x} = c_{1} e^{x} - c_{2} e^{- x} \frac{d^{2} y}{d x^{2}} = c_{1} e^{x} - c_{2} e^{- x} = y \frac{d^{2} y}{d x^{2}} - y = 0$