MCQ
Which of the following differential equations has $y=c_{1} e^{x}+c_{2} e^{-x}$ as the general solution?
- ✓$\frac{d^{2} y}{d x^{2}}-y=0$
- B$\frac{d^{2} y}{d x^{2}}+y=0$
- C$\frac{d^{2} y}{d x^{2}}+1=0$
- D$\frac{d^{2} y}{d x^{2}}-1=0$
$y=c_{1} e^{x}+c_{2} e^{x}$ .........$(1)$
Differentiating with respect to $\mathrm{x}$, we get:
$\frac{d y}{d x}=c_{1} e^{x}-c_{2} e^{-x}$
Again, differentiating with respect to $\mathrm{x}$, 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 $\mathrm{A}$.
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