88 Laplace transform of a differential equation

Suppose that $y(0)=p_0,y^{\prime }(0)=p_1,\mathrm{\dots },y^{(n-1)}(0)=p_{n-1}$ and

$$y^{(n)}(t)+a_{n-1}{y}^{(n-1)}(t)+\mathrm{\dots }+a_{0}y=u(t)$$

where the coefficients are complex constants and $u$ satisfies $(E)$. Then there exists a complex polynomial $q_{n-1}(s)$ of degree $\le n-1$ such that

$$(s^n+a_{n-1}{s}^{n-1}+\mathrm{\dots }+a_0)ℒ(y)(s)+q_{n-1}(s)=ℒ(u)(s).$$

The characteristic equation of the differential equation is

$$s^n+a_{n-1}{s}^{n-1}+\mathrm{\dots }+a_0=0.$$