Thread: differential equation to algebraic form

using substitution x = e^(kt) convert the following differential equation into an algebraic equation for k (d^2)x/dt^2 + 2B(dx/dt) + (w^2)x = 0 with w>0 after substituting for x what must i do to convert to algebraic form i was not taught anything like this in lectures

Hello, crafty!

This is one of the classic methods for Differential Equations.
It should have been demonstrated to you in a lecture.

Using substitution $\displaystyle x = e^{kt}$, convert the following differential equation
into an algebraic equation for $\displaystyle k\!:\;\;\frac{d^2x}{dt^2} + 2B\frac{dx}{dt} + w^2x \:= \:0\;\;\text{with }w>0$

After substituting for $\displaystyle x$, what must i do to convert to algebraic form?
i was not taught anything like this in lectures.

We have: .$\displaystyle \frac{d^2x}{dt^2} + 2B\frac{dx}{dt} + w^2x \:=\:0$ .[1]

. . And: .$\displaystyle \begin{Bmatrix} x &=& e^{kt} \\ \\[-4mm]\dfrac{dx}{dt} &=& ke^{kt} \\ \\[-4mm] \dfrac{d^2x}{dt^2} &=& k^2e^{kt} \end{Bmatrix}$


Substitute into [1]: .$\displaystyle k^2e^{kt} + 2B\!\cdot\!ke^{kt} + w^2\!\cdot\!e^{kt} \;=\;0 $


Divide by $\displaystyle e^{kt}\!:\;\;k^2 + 2Bk + w^2 \;=\;0$ . (This is the "algebraic form".)

thanks