Differential Equations

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

Linear equation, nonlinear equation

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

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

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

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

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

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

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)

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.

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

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

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:

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

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

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

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)

Nonhomogeneous linear equations Method of variation of parameters: Example: