A spring/mass/dashpot system has mass 1 kg, damping constant 7 kg/sec and spring constant 12 kg per sq sec. The system starts at rest and then has an external force of e7t\displaystyle {e}^{{-{7}{t}}} Newtons applied after t\displaystyle {t} seconds. The IVP below models the system:

x+7x+12x=e7t\displaystyle {x}{''}+{7}{x}'+{12}{x}={e}^{{-{7}{t}}}, x(0)=0,x(0)=0\displaystyle \quad{x}{\left({0}\right)}={0},\quad{x}'{\left({0}\right)}={0}

The Laplace transform of the IVP has solution Y(s)=F(s)Φ(s)\displaystyle {Y}{\left({s}\right)}={F}{\left({s}\right)}\cdot\Phi{\left({s}\right)} where F(s)\displaystyle {F}{\left({s}\right)} represents the Laplace transform of the forcing term e7t\displaystyle {e}^{{-{7}{t}}} and Φ(s)\displaystyle \Phi{\left({s}\right)} represents the transfer function.

Φ(s)=\displaystyle \Phi{\left({s}\right)}=  

The weight function w(t)\displaystyle {w}{\left({t}\right)} is the inverse Laplace transform of the transfer function.

w(t)=L1(Φ(s))=\displaystyle {w}{\left({t}\right)}={L}^{{-{1}}}{\left(\Phi{\left({s}\right)}\right)}=  

The solution to the IVP is the convolution of the forcing term with the weight function.

f(t)*w(t)=\displaystyle {f{{\left({t}\right)}}}\text{*}{w}{\left({t}\right)}=