42 Case Study 7: Approximating Series Solution to an ODE Available to Purchase

A linear second-order, initial value problem (IVP), ordinary differential equation (ODE) is A(d²y/dx²) + B(dy/dx) + Cy = u(x). The initial conditions are y(x = x₀) = y₀ and $dy/dx|x=x0=y˙0$⁠. In an ideal case the forcing function is a constant, held steady, for all x after the beginning at x₀, u(x ≥ x₀) = u_SS. If A = 0 it is a first-order ODE, not second-order. For A ≠ 0, the analytical solution for this ODE is relatively simple (if one is practiced in solving ODEs) and results in a model of the form $y(x≥x0)=α+βe−(x−x0)/τ1+γe−(x−x0)/τ2$⁠. There are three cases. In the case in which B² > 4AC, the process is monotonic and asymptotically stable with τ-values $τ1=2A/(B+B2−4AC)$ and $τ2=2A/(B−B2−4AC)$⁠. Then α = u_SS/C, $β=[τ1τ2y˙0+τ1(y0−α)]/(τ1−τ2)$⁠, and $γ=−[τ1τ2y˙0+τ2(y0−α)]/(τ1−τ2)$⁠.