1. Differential Equations

2. Ordinary differential equation (ODE) , Partial differential equation (PDE)
Order : highest derivative in equation

3. Linear equation, nonlinear equation

4. Differential equations
Ordinary differential equation (ODE) , Partial differential equation (PDE)
Order : highest derivative in equation
Linear equation, nonlinear equation

5. Differential equations
Homogeneous equation, nonhomogeneous equation
Implicit solution, explicit solution

6. Differential equations
General solution, particular solution
Initial value problem, boundary value problem
Exact, approximate, and numerical solutions

7. First order differential equation, I
Separable equations;
Equations reducible to separable form;

8. First order differential equation, I
Exact differential equations;
Integrating factor; if ODE is not exact, it can be made exact.

9. First order differential equation, II
Linear first-order differential equation (homogeneous);
Linear first-order differential equation (nonhomogeneous);

10. First order differential equation, II
Method of variation of parameters
General solution of homogeneous eq. is given by
Idea is that we may replace the integration constant c by u(x)

11. Ordinary Linear differential equation, I
Second order linear differential equation,
Theorem (Superposition Principle): Any linear combination of solutions of the homogeneous linear differential equation is also a solution.

12. Ordinary Linear differential equation, I
Homogeneous 2nd order ODE with constant cocefficients
Try
Characteristic equation (or auxiliary equation)
Roots:
Solutions:
Examples:

13. Two functions are linearly dependent if they are proportional.
Ordinary Linear differential equation, II
Two functions are linearly dependent if they are proportional.
Theorem: If y1 and y2 are linearly independent solutions of ODE, the general solution of ODE is
If 1 2, general solution is
If 1, 2 are complex conjugate, the solutions are complex

14. Real solutions from these complex solutions by Euler formulas:
Ordinary Linear differential equation, II
Real solutions from these complex solutions by Euler formulas:
Corresponding general solution:
Example: initial value problem:

15. Ordinary Linear differential equation, III
Double root case (critical case)
Second solution by method of variation of parameters
Corresponding general solution:

16. Ordinary Linear differential equation, III
Example:
Summary: For the equation
Case
Roots
General solution I
Distinct real 1, 2 II
Complex conjugate
1=p+iq, 2=p-iq
III
Real double root =-a/2

17. Ordinary Linear differential equation, IV
Cauchy equation( or Euler equation)
Try y = xm
The general solution is
Example:
Critical case:

18. Nonhomogeneous linear equations
Theorem: A general solution y(x) of the linear nonhomogeneous differential equation
is the sum of a general solution yh(x) of the corresponding homogeneous equation and an arbitrary particular solution yp(x)

19. Nonhomogeneous linear equations
Method of variation of parameters:
Example: