Second Order Linear D.E. Summary
Solve the Characteristic Equation: .
Case 1) Two real roots :
Case 2) Real double root :
Case 3) Complex Conjugate root , where :
General solution of non-homogenous D.E.: where is the general solution of the homogenous equation, and is the particular solution (with no arbitrary constants).
Method of Undetermined Coefficients (Guess and try method)
Only works if is polynomial, exponential, sine or cosine (or sum/product of these).
Polynomial: Try =Polynomial (e.g. or .)
Exponential (): Try , where is a function of .
Trigonometric ( or ): Convert to complex differential equation by replacing with , replace / by .
Try , where is a function of . After solving for , take real/imaginary part of for cosine/sine respectively.
Method of variation of parameters
[Step 1)] Solve the homogenous D.E. .
Get solution of the form .
Let and where is the Wronskian
Particular solution: .
General solution: .
Let be the amplitude of the driving (external) force. If , by Newton’s Second Law, , hence
where . The value is called the natural frequency.
If , then
where is the driving (external) frequency.
At resonance (when ),