6.2: General Solution of the Standard Stable First Oder ODE and IC by Application of the Convolution Integral

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From Equation 3.4.8, we have for stable 1^st order systems:

\[\mathrm{ODE}+\mathrm{IC}: \dot{x}+\left(1 / \tau_{1}\right) x=b u(t), \quad x(0)=x_{0}, \quad \text { find } x(t) \text { for } t>0\label{eqn:6.4} \]

Solve by first taking the Laplace transform,

\[L[\mathrm{ODE}]: \quad s X(s)-x_{0}+\left(1 / \tau_{1}\right) X(s)=b U(s) \nonumber \]

\[\Rightarrow \quad X(s)=\frac{x_{0}}{s+1 / \tau_{1}}+b\left(\frac{1}{s+1 / \tau_{1}}\right) \stackrel{F_{2}(s)}{U(s)} \nonumber \]

Take the inverse transform, using Equation 2.2.6 and convolution integral Equation 6.1.5:

\[x(t)=x_{0} e^{-t / \tau_{1}}+b \int_{\tau=0}^{\tau=t} e^{-t / \tau} u(t-\tau) d \tau=x_{0} e^{-t / \tau_{1}}+b \int_{t=0}^{\tau=t} e^{-(t-\tau) / \tau_{1}} u(\tau) d \tau\label{eqn:6.5} \]

Equations \(\ref{eqn:6.5}\) are general solutions of problem Equation \(\ref{eqn:6.4}\) applicable for any input \(u(t)\)¹. For both forms, the first term is obviously the initial-condition (IC) response and the second term is the forced response. In applications with specific \(u(t)\) functions, the second form of the forced-response integral on the right-hand side of Equations \(\ref{eqn:6.5}\) is used more commonly than the first. The first form is also valid, but the functional nature of \(u(t-\tau)\) can sometimes be difficult to interpret correctly.

The forced-response integrals in Equations \(\ref{eqn:6.5}\) are called convolution integrals, as in Equation 6.1.3. They are also sometimes known as superposition integrals, because, as is shown in Section 8.10, they can be derived as the linear superposition of responses to differentially small inputs.

In Equations \(\ref{eqn:6.5}\), constants \(\tau_1\) and \(b\) should be expressed in terms of the physical constants of the actual system analyzed, such as mass \(m\), damping constant \(c\), etc.

¹Note, however, that solutions Equations \(\ref{eqn:6.5}\) are not valid for a non-standard 1^st order ODE, e.g., one with right-hand-side dynamics such as Equation 5.4.2.