differential equation to algebraic form

**crafty** · February 4th 2009, 09:37 AM

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

**Soroban** · February 4th 2009, 12:31 PM

Hello, crafty!

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

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

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

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

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

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

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

**crafty** · February 4th 2009, 02:35 PM

thanks

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