# Which of the following differential equations has

Question:

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

Solution:

The given equation is:

$y=c_{1} e^{x}+c_{2} e^{-x}$           ...(1)

Differentiating with respect to x, we get:

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

Again, differentiating with respect to x, we get:

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

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

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

This is the required differential equation of the given equation of curve.

Hence, the correct answer is B.