## Second Order Linear D.E. Summary

**Homogenous D.E.**

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Solve the *Characteristic Equation*: .

Case 1) Two real roots :

Case 2) Real double root :

Case 3) Complex Conjugate root , where :

**Non-homogenous D.E.**

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)**

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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**

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[Step 1)] Solve the homogenous D.E. .

Get solution of the form .

[Step 2)]

Let and where is the Wronskian

Particular solution: .

General solution: .

**Forced Oscillations**

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 ),