Which of the following differential equations | vedclass

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(A) The given equation is:$y=c_{1} e^{x}+c_{2} e^{-x}$  $(1)$Differentiating with respect to $x$,we get:$\frac{d y}{d x}=c_{1} e^{x}-c_{2} e^{-x}$Agai

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

Vedclass · Answer

(A) 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:$\frac{d^{2} y}{d x^{2}}=c_{1} e^{x}+c_{2} e^{-x}$Since the right-hand side is equal to $y$ from equation $(1)$,we have:$\frac{d^{2} y}{d x^{2}}=y$Rearranging the terms,we get:$\frac{d^{2} y}{d x^{2}}-y=0$This is the required differential equation.Hence,the correct answer is $A$.

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

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