y=A e^x+B e^{-2 x} satisfies which of the fol | vedclass

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(D) Given the equation $y=A e^x+B e^{-2 x}$.First,differentiate with respect to $x$:$\frac{d y}{d x} = A e^x - 2B e^{-2x}$.Second,differentiate again

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

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(D) Given the equation $y=A e^x+B e^{-2 x}$.First,differentiate with respect to $x$:$\frac{d y}{d x} = A e^x - 2B e^{-2x}$.Second,differentiate again with respect to $x$:$\frac{d^2 y}{d x^2} = A e^x + 4B e^{-2x}$.Now,we eliminate the constants $A$ and $B$. From the first derivative,$A e^x = \frac{d y}{d x} + 2B e^{-2x}$.Substitute this into the second derivative equation:$\frac{d^2 y}{d x^2} = (\frac{d y}{d x} + 2B e^{-2x}) + 4B e^{-2x} = \frac{d y}{d x} + 6B e^{-2x}$.Alternatively,consider the characteristic equation for the roots $m_1 = 1$ and $m_2 = -2$. The differential equation is $(D-1)(D+2)y = 0$,where $D = \frac{d}{dx}$.Expanding this: $(D^2 + 2D - D - 2)y = 0$,which simplifies to $\frac{d^2 y}{d x^2} + \frac{d y}{d x} - 2y = 0$.

$y=A e^x+B e^{-2 x}$ satisfies which of the following differential equations?

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