# Nonhomogeneous ODE with Dirac delta

1. Apr 25, 2008

### grilo

Trying to solve the ODE
$$mx''(t) + bx'(t) + kx(t) = F(t)$$
with $$m$$ measured in $$Kg$$, $$b$$ in $$Kg/s$$ and $$Kg/s^2$$, $$F(t)$$ in $$Kgm/s^2$$ and $$x(t)$$ in $$m$$ with initial conditions $$x(0) = 0$$ and $$x'(0) = 0$$, i got the following Green's function

$$G(t,t') = \frac{1}{m\omega} e^{-\omega_1(t-t')}\sinh\left[\omega(t-t')\right]$$
for $$t \geq t'$$, with
$$\omega=\sqrt{\omega_1^2+\omega_0^2}$$
$$\omega = \sqrt{\frac{k}{m}}$$
$$\omega_1=\frac{b}{2m}$$.

Adding up all the responses to unit impulses i got

$$x(t) = \int_0^t F(t')G(t,t')dt'$$

So far, everything was ok.

But when i tried to get the response of the ODE for a unit impulse of amplitude $$f_0$$ at $$t = t_0$$ ($$F(t) = f_0\delta(t-t_0)$$) i integrated the equation above but the dimension of the expression i got for $$x(t)$$ is $$m/s$$ instead of $$m$$.

If $$F(t)$$ is any other function it works fine, but with $$F(t) = f_0\delta(t-t_0)$$ I get this dimensional error.