What would you like to understand?

The characteristic equation is the quadratic algebraic equation $0 = ar^2 + br + c$. It is obtained by substituting the trial solution $y = e^{rt}$ into the second-order linear homogeneous differential equation $ay'' + by' + cy = 0$, factoring out the nonzero exponential term $e^{rt}$, and setting the remaining polynomial in $r$ to zero.

Conditions: The differential equation is $ay'' + by' + cy = 0$.; The trial solution is $y = e^{rt}$.; $a, b, c$ are constants.

The characteristic equation is the quadratic algebraic equation $0 = ar^2 + br + c$. It is obtained by substituting the trial solution $y = e^{rt}$ into the second-order linear homogeneous differential equation $ay'' + by' + cy = 0$, factoring out the nonzero exponential term $e^{rt}$, and setting the remaining polynomial in $r$ to zero.

Conditions: The differential equation is $ay'' + by' + cy = 0$.; The trial solution is $y = e^{rt}$.; $a, b, c$ are constants.