Correct Answer - Option 2 :
\(\rm \frac{d^2y}{dx^2}-y=0\)
Concept:
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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.