Constant Coefficient Linear ODEs

1
Constant Coefficient Linear ODEs

• Second-order ODEs
• Homogeneous case
• Non-homogeneous case
• Higher-order ODEs

2
Linear ODEs

• Second-order
• Constant coefficients
• Homogeneous
• Higher order

3
Existence and Uniqueness of Solutions

• Second-order initial value problem (IVP)
• Existence and uniqueness
• Assume that p(x) q(x) are continuous functions
  on an interval I and x0 is contained in I, then
  the IVP has a unique solution y(x) on the
  interval I
• The general solution has the form y(x)
  c1y1(x)c2y2(x)
• Linear independence
• Two solutions y1(x) and y2(x) are linearly
  independent if and only if their Wronskian
  W(y1,y2) is non-zero at some point x contained in
  I

4
Second-Order Homogeneous ODE

• General form
• Proposed solution
• Characteristic equation

5
Three Possible Cases

• Two distinct real roots
• One repeated real root
• Need second independent solution
• Two complex conjugate roots
• Need two real solutions

6
Mixing Tank Example

• Mass balance on each tank

7
Model Reformulation

• From second ODE
• Substitute into first ODE simplify
• Substitute parameter values Fin(t) 10 m3/h
• Refine dependent variable to obtain homogeneous
  equation

8
Model Solution

• Homogeneous equation
• Solution form
• Evaluate constants
• Solution for original equation

9
Higher-Order Homogeneous ODE

• General form
• Characteristic equation
• Solution basis
• The solutions y1(x) el1x, , yn(x) elnx form
  a basis if and only if the l1,, ln are distinct
• Distinct real roots

10
Other Cases

• Simple complex roots
• Real roots repeated m times
• Twice repeated complex root

11
Second-Order Nonhomogeneous ODE

• General form
• General solution
• yh(x) is the solution of homogeneous ODE with
  r(x) 0
• yp(x) is a particular solution of the
  nonhomogeneous ODE
• Method of undetermined coefficients
• Use form of r(x) to determine proposed form of
  yp(x)
• Substitute yp(x) into ODE solve for the unknown
  coefficients in yp(x)

12
Method of Undetermined Coefficients

• Selection of particular solution form
• If r(x) appears in Table 2.1, select
  corresponding yp(x)
• If r(x) is the sum of entries in Table 2.1,
  select yp(x) to be the sum of the corresponding
  entries
• If a term in yp(x) is a solution of the
  homogeneous ODE, multiple the yp(x) in Table 2.1
  by x

13
Mixing Tank Example cont.

• Fin(t) 10e-t m3/h
• Nonhomogeneous equation
• Particular solution
• Complete solution

14
Higher-Order Nonhomogeneous ODE

• General form
• General solution
• yh(x) is the solution of homogeneous ODE with
  r(x) 0
• yp(x) is a particular solution of the
  nonhomogeneous ODE
• Method of undetermined coefficients
• Follow same rules as before
• If a term in yp(x) is a solution of the
  homogeneous ODE, multiple the yp(x) in Table 2.1
  by the lowest power xk such that no term of
  xkyp(x) is a solution of the homogeneous equation