The differential equation corresponding to th | vedclass

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Question

(A) Given the family of curves: $y = a + b e^{2x} + c e^{-3x}$ $(i)$ Differentiating with respect to $x$: $y_1 = 2b e^{2x} - 3c e^{-3x}$ (ii) Differen

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$y_3+y_2-6 y_1=0$

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(A) Given the family of curves: $y = a + b e^{2x} + c e^{-3x}$ $(i)$ Differentiating with respect to $x$: $y_1 = 2b e^{2x} - 3c e^{-3x}$ (ii) Differentiating again: $y_2 = 4b e^{2x} + 9c e^{-3x}$ (iii) Differentiating a third time: $y_3 = 8b e^{2x} - 27c e^{-3x}$ (iv) To eliminate the constants $a, b, c$, we consider the linear combination $y_3 + k_1 y_2 + k_2 y_1 = 0$. The characteristic equation for the terms $e^{2x}$ and $e^{-3x}$ is $(m-2)(m+3)m = 0$, which is $m(m^2 + m - 6) = 0$, or $m^3 + m^2 - 6m = 0$. This corresponds to the differential equation $y_3 + y_2 - 6y_1 = 0$.

The differential equation corresponding to the family of curves given by $y=a+b e^{2 x}+c e^{-3 x}$ is

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