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Case Study 7: Approximating Series Solution to an ODE

Excerpt

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).

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
42.2Exercises
42.1Concepts and Analysis
42.2Exercises

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