Question

The differential equation of the family of curves $y=c_1{e}^x+c_2{e}^{-x}$ is:

• (A) $\frac{d^{2}y}{{dx}^2}+y=0$
• (B) $\frac{d^{2}y}{{dx}^2}-y=0$
• (C) $\frac{d^{3}y}{d{x}^3}+y=0$
• (D) $\frac{d^{4}y}{{dx}^4}-1=0$

Answer 1

The Correct Option is B

Explanation:
Given,$y=c_1{e}^x+c_2{e}^{-x}$Differentiating w.r.t. $x$We get,$\frac{dy}{dx}=\frac{d}{dx}{c}_1{e}^x+\frac{d}{dx}{c}_2{e}^{-x}$$\Rightarrow \frac{dy}{dx}=c_1{e}^x-c_2{e}^{-x}$$\Rightarrow \frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}(c_1{e}^x-c_2{e}^{-x})$$\Rightarrow \frac{d^{2}y}{d{x}^2}=c_1{e}^x+c_2{e}^{-x}=y$$\Rightarrow \frac{d^{2}y}{d{x}^2}-y=0$Hence, the correct option is (B).