General Linear ODEs

1
General Linear ODEs

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

2
Linear ODEs

• Second-order ODE with constant coefficients
• General second-order ODE
• Homogeneous ODE
• Higher order ODE

3
Homogeneous Second-Order ODEs

• Homogeneous second-order linear ODE
• Linear independence
• Two solutions y1(x) y2(x) are linearly
  independent if
• General solution
• A general solution has the form y(x)
  c1y1(x)c2y2(x) where the basis y1(x) y2(x) are
  linearly independent the constants c1 and c2
  are arbitrary
• The complete solution is obtained by using the
  initial conditions to determine the constants

4
Existence and Uniqueness of Solutions

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

5
Reduction of Order

• Assume one solution is known y1(x)
• Define second solution y2(x) u(x)y1(x)
• Substitute y2(x) into ODE generate first-order
  ODE for U(x) du/dx
• Solve first-order ODE for U(x)
• Integrate U(x) to find second solution

6
Euler-Cauchy Equations

• Solution of second-order ODEs with non-constant
  coefficients is generally non-trivial
• Explicit solution formulas available for
  Euler-Cauchy equations
• Solution form auxiliary equation
• Roots m1, m2

7
Three Cases

• Distinct real roots
• Solutions y xm are linearly independent
• General solution
• Repeated real roots
• Solutions y xm are not linearly independent
• General solution
• Complex roots
• Not practically important
• See text for example

8
Euler-Cauchy Equation Example

• ODE
• Substitute solution y(x) xm
• Find roots
• Solution
• Evaluate constants

9
Homogeneous Higher-Order ODE

• Homogeneous nth-order linear ODE
• Existence and uniqueness
• Assume that p0(x), p1(x),, pn-1(x) are
  continuous functions on an open interval I x0
  is contained in I, then the IVP has a unique
  solution y(x) on the interval I
• General solution
• Form y(x) c1y1(x)c2y2(x) cnyn(x)
• The complete solution is obtained by using the
  initial conditions to determine the constants

10
Linear Independence of Solutions

• The basis y1(x), y2(x),, yn(x) must be linearly
  independent
• Wronskian
• The solutions y1(x), y2(x),, yn(x) are linearly
  independent if and only if their Wronskian W is
  non-zero at some point x contained in I

11
Nonhomogeneous Second-Order ODE

• Nonhomogeneous second-order linear ODE
• General solution
• yh(x) is the solution of homogeneous ODE with
  r(x) 0
• yp(x) is a particular solution of the
  nonhomogeneous ODE
• Obtaining particular solutions
• Method of undetermined coefficients useful if
  the derivatives of r(x) have the same general
  form as r(x)
• Variation of parameters more general but more
  complex

12
Variation of Parameters

• Particular solution
• y1(x) y2(x) are a solution basis for the
  homogeneous ODE
• W(x) is the Wronskian
• Integrals may be difficult or impossible to
  evaluate

13
Nonhomogeneous Higher-Order ODE

• Nonhomogeneous ODE with variable coefficients
• General solution
• yh(x) is the solution of homogeneous ODE with
  r(x) 0
• yp(x) is a particular solution of the
  nonhomogeneous ODE
• Particular solution
• Wronskians W1(x),,Wn(x) defined in text