ECE602 BME I

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• ECE602 BME I
• Ordinary Differential Equations in Biomedical
  Engineering

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• Classification of ODEs
• Canonical Form of ODE
• Linear ODEs
• Nonlinear ODEs
• Steady-State Solutions and Stability Analysis
• BME Example 1- The dynamics of Drug Absorption
• BME example 2 Hodgkin-Huxley Model for Dynamics
  of Nerve Cell Potentials

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Classification of ODEs

• General Form of ODE

• The order of an ODE the order of the highest
  derivative
• R(t)0 Homogeneous R(t)?0 Nonhomogeneous
• Nonlinear an ODE contains powers of the
  dependent variable, powers of the derivatives,
  or products of the dependent variable with the
  derivatives

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Classification of ODEs

• Examples

First-order, linear, homogeneous
First-order, linear, nonhomogeneous
First-order, nonlinear, nonhomogeneous
Second-order, linear, nonhomogeneous
Second-order, nonlinear, nonhomogeneous
Third-order, nonlinear, nonhomogeneous
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Canonical Form of ODEs

• Canonical form
• A set of n simultaneous first-order ODEs
• Required for methods for integrating ODEs

Vector format
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Canonical Form of ODEs
Transformation to Canonical form
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Linear ODEs
Matrix Exponential Method
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Linear ODEs
EXPM Matrix exponential. EXPM(A) is the
matrix exponential of A.
gtgt syms t gtgt gtgt A1 1-1 1 y011 gtgt
yexpm(At)y0 y exp(t)cos(t)exp(t)sin(t)
-exp(t)sin(t)exp(t)cos(t)
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Linear ODEs
Eigenvector matrix
Eigenvalue matrix
Method using eigenvalues and eigenvectors
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Linear ODEs
EIG Eigenvalues and eigenvectors. X,D
EIG(A) produces a diagonal matrix D of
eigenvalues and a full matrix X whose columns
are the corresponding eigenvectors so that
AX XD.
gtgt syms t gtgt A1 1-1 1 y011 gtgt
X,Deig(A) gtgt y(Xexpm(Dt)inv(X))y0 y
exp(t)cos(t)-1/2i(exp(t)cos(t)iexp(t)sin(
t))1/2i(exp(t)cos(t)-iexp(t)sin(t))
1/2i(exp(t)cos(t)iexp(t)sin(t))-1/2i(exp(t
)cos(t)-iexp(t)sin(t))exp(t)cos(t) gtgt
ysimplify(y) y exp(t)(cos(t)sin(t))
exp(t)(-sin(t)cos(t))