# Which one of the following differential equations has the general solution y = aex + be-x?

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Which one of the following differential equations has the general solution

y = aex + be-x?

1. $\rm \frac{d^2y}{dx^2}+y=0$
2. $\rm \frac{d^2y}{dx^2}-y=0$
3. $\rm \frac{d^2y}{dx^2}+y=1$
4. $\rm \frac{dy}{dx}-y=0$

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Correct Answer - Option 2 : $\rm \frac{d^2y}{dx^2}-y=0$

Concept:

• Differential Equation: A differential equation is an equation that relates one or more functions and their derivatives.
• e.g. $\rm \frac{dy}{dx}$ + x = 2y + 3, etc.
• $\rm \dfrac{d}{dx}e^{f(x)}=\dfrac{d}{dx}f(x)e^{f(x)}$

Calculation:

y = aex + be-x

⇒ $\rm \frac{dy}{dx}=\frac{d}{dx}ae^x +\frac{d}{dx}be^{-x}$

⇒ $\rm \frac{dy}{dx}$ = aex - be-x

⇒ $\rm \frac{d}{dx}\left(\frac{dy}{dx}\right)=\frac{d}{dx}(ae^x -be^{-x})$

⇒ $\rm \frac{d^2y}{dx^2}$ = aex + be-x = y

∴ The general solution of y = aex + be-x is $\rm \frac{d^2y}{dx^2}$ - y = 0.