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.