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Integrating Factor

Show that y...

Question

Show that $y = c_{1} e^{x} + c_{2} e^{- x}$ is the general solution of the differential equation $\frac{d^{2} y}{d x^{2}} - y = 0$ .

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Solution

Differentiate on both sides

$\Rightarrow \frac{d y}{d x} = c_{1} e^{x} - c_{2} e^{- x}$

Differentiate again on both sides

$\Rightarrow \frac{d^{2} y}{d x^{2}} = c_{1} e^{x} + c_{2} e^{- x}$

$\Rightarrow \frac{d^{2} y}{d x^{2}} = y$

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

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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. Which of the following differential equation has

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

as the general solution?

y = A cos n x + B sin n x

, is a solution of the differential equation:

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

Q. The complete solution of the ordinary differential equation

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

y = c_{1} e^{- x} + c_{2} e^{- 3 x}

p

q

Q. The complete solution of the ordinary differential equation

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

y = c_{1} e^{- x} + c_{2} e^{- 3 x}

Which of the following is a solution of the differential equation

\frac{d^{2} y}{d x^{2}} + p \frac{d y}{d x} + (q + 1) y = 0

?

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