Case Study 7: Approximating Series Solution to an ODE


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(xx0) = uSS. 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(xx0)=α+βe(xx0)/τ1+γe(xx0)/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+B24AC) and τ2=2A/(BB24AC). Then α = uSS/C, β=[τ1τ2y˙0+τ1(y0α)]/(τ1τ2), and γ=[τ1τ2y˙0+τ2(y0α)]/(τ1τ2).

42.1Concepts and Analysis

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In