# 1.3: Solving First-Order ODE Models

The response of a first-order ODE to a step forcing function with given initial conditions can be obtained with the help of the Laplace Transform. Accordingly, we consider the generic first-order ODE model: $$\tau \frac{dy\left(t\right)}{dt}+y\left(t\right)=u(t)$$, and apply Laplace transform assuming an initial condition $$y\left(0\right)=y_0$$, to obtain: $\tau \left(sy\left(s\right)-y_0\right)+y(s)=u(s)$ Next, assuming a unit step input $$u\left(t\right)$$, where $$u\left(s\right)=\frac{1}{s}$$, the output is solved as: $y\left(s\right)=\frac{1}{s\left(\tau s+1\right)}+\frac{\tau y_0}{\tau s+1}$ We may use partial fraction expansion (PFE) to express the output as: $y\left(s\right)=\frac{1}{s}-\frac{\tau }{\tau s+1}+\frac{\tau y_0}{\tau s+1}$ After applying the inverse Laplace transform, we obtain the time-domain solution to the ODE as: $y\left(t\right)=\left[1+\left(y_0-1\right)e^{-t/\tau }\right]u\left(t\right)$ The $$u\left(t\right)$$ in the above expression represents a unit step function, that is used to represent causality, i.e., the output is valid for $$t\ge 0$$.

The steady-state value of the system response is denoted as: $$y_{\infty }={\mathop{\mathrm{lim}}_{t\to \infty } y(t)\ }$$. In terms of the steady-state output, the step response of the first-order ODE model is given as: $y\left(t\right)=\left[y_{\infty }+\left(y_0-y_{\infty }\right)e^{-t/\tau }\right]u\left(t\right)$ Assuming zero initial conditions, i.e., $$y_0=0$$, the output of the system is expressed as: $y\left(t\right)=\left(1-e^{-t/\tau }\right)u\left(t\right)$ The output at the selected times, $$t=k\tau ,\ \ k=0,1,\dots$$ are compiled in the following table:

Time Output value
0 $$y\left(0\right)=0$$
1$$\tau$$ $$1-e^{-1}\cong 0.632$$
2$$\tau$$ $$1-e^{-2}\cong 0.865$$
3$$\tau$$ $$1-e^{-3}\cong 0.950$$
4$$\tau$$ $$1-e^{-4}\cong 0.982$$
5$$\tau$$ $$1-e^{-5}\cong 0.993$$

image3 By convention, the model output is assumed to have reached steady-state when the output attains 98% of its final value. Hence, the settling time of the system is given as: $$t_s=4\tau$$. The output for $$\tau =1sec$$ is plotted in Figure 6.

Figure 6: Output of a first-order system model to a unit-step input.