CPT 02: ODEs

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Differential Equations

Definition: Differential Equation (DE)

Differential Equation (DE): A model (i.e., equation) that contains derivatives (e.g., rate of change, acceleration, etc.) of an unknown function.

Definition: Ordinary Differential Equation (ODE)

Ordinary Differential Equation (ODE): A DE that contains only ordinary derivatives (i.e., the unknown function is a function of a single independent variable), usually written as dydx=f(x,y).

Definition: Partial Differential Equation (PDE)

Partial Differential Equation (PDE): A DE that contains only partial derivatives (i.e., the unknown function is a function of more than one independent variable).

An example of a PDE is Fick's second law: u(t,x,y)t=D(2u(t,x,y)x2+2u(t,x,y)y2).

NOTE: This course will only consider ODEs.

Definition: Order of a DE

Order: The order of a DE is the order of the highest-order derivative.

Definition: Linear DE

Linear DE: A DE is linear if

  1. the coefficients of the derivatives (including zeroth derivative) are functions of only the independent variable, and

  2. all of the derivatives (including zeroth derivative) are only to the first power.

If either of the conditions is not met, then the DE is nonlinear.

Question 01

Identify the dependent variable, independent variable(s), and parameters in the following DEs:

  1. Exponential growth and decay: dxdt=kx

  2. The relation between air pressure, p, and altitude, y: dpdy=ρ(y)g

  3. RLC circuit: Ld2Idt2+RdIdt+1CI=0

  4. Black-Scholes equation: 12σ2S22VS2+rSVS+VtrV=0

  5. Newton's Second Law: Fnet=md2xdt2

  6. Axial deformation on a bar: ddx[A(x)σ(x)]+f(x)=0

  7. Differential equations to explore elastic beams: d2dx2(EId2νdx2)+kν=q

  8. Torsion of elastic bars: 2θx2+2θy2+2=0

  9. Irrotational fluid flow: 2ϕx2+2ϕy2=0

  10. DEs to explore cancer-immune dynamics and tumor dormancy: dCdt=μα(1+Ψ(I,C))C(1(CKc)α)

  11. Free fall with air resistance: mdvdt=mg12ρCDAv2

  12. Decomposition of NO2: d[NO2]dt=k[NO2]2

  13. Torricelli's Law: dVdt=kV

  14. Black hole mass: dMdt=kM2

  15. Solow-Swan model: kt=skαδk

  16. Bertalanffy equation: dWdt=ηSkV

  17. Wave equation in one-dimension: 2ut2=c22ux2

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Last Modified: Sunday, 23 August 2020 20:40 EDT