An ODE with boundary conditions at infinity

Question

I have a problem where:

$\ddddot{x} - 2 \ddot{x} + x = 0$

With boundary conditions

$x(0) = 1, \dot{x}(0) = 2, x(\infty) = 0, \dot{x}(\infty) = 0$

So I get my characteristic equation:

$s^4 -2s^2 + 1 = 0$

$s = 1,1,-1,-1$

I have to add a couple extra t terms to my general solution due to the repeated roots:

$x(t) = A_1e^t + A_2te^t + A_3e^{-t} + A_4te^{-t}$

To setup my simultaneous equations to solve for the coefficients, I need to get the derivative:

$\dot{x}(t) = A_1e^t + A_2(e^t + te^t) - A_3e^{-t} + A_4(e^{-t} - te^{-t})$

So I can quickly setup my equations using t=0:

$x(0) = 1 = A_1 + A_3$
$\dot{x}(0) = 2 = A_1 + A_2 - A_3 + A_4$

but for the infinity terms, I'm getting equations with infinity in them...not sure what to do. Any suggestions?

Answer 1

Note that $t\mathrm{e}^t$ is asymptotically larger than the other solutions, and tends to $\infty$ as $t$ tends to $\infty$. Hence $A_2=0$. Similarly $A_1=0$, as $\mathrm{e}^t\to\infty$, while the rest of the terms tend to 0.

Hence $$ x(t)=(at+b)\mathrm{e}^{-t}. $$ The constants $a,b$ can be found from the initial data.