3.1-3.3 Constant Coefficients

3.1-3.3 Constant Coefficients - GE 207K Summary 2nd Order...

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GE 207K 2nd Order ODEs with Constant Coe ff s September 13, 2011 Summary : We are dealing with 2nd-order linear homogeneous ordinary di ff erential equations with con- stant coe cients (note all the characteristics associated – pay special attention to the term constant coe cients), whose standard form is given by y + Ay + By = 0 , (1) where A and B are two CONSTANTS . We solve these di ff erential equations by saying that our solutions appear in the form of y = e rt , where r is a constant . By requiring that y = e rt be a solution, we end up with the corresponding characteristic equation for the ODE: r 2 + Ar + B = 0 . (2) Recall from algebra that Eq. ( 2 ) is a quadratic equation with three possible solution types: Real and distinct roots – i.e. r 1 , r 2 R and r 1 = r 2 . Real and repeated roots – i.e. r 1 = r 2 R . Pair of complex conjugate numbers – i.e. r 1 = λ + , and r 2 = ¯ r 1 = λ . Steps : 1. Write down the corresponding characteristic equation to Eq. ( 1 ). r 2 + Ar + B = 0 (3) 2. Solve for the roots. Note that you need to find two roots. The general solution depends on the form of the roots: Real and distinct roots – i.e. r 1 , r 2 R , r 1 = r 2 . Then general solution is y = c 1 e r 1 t + c 2 e r 2 t . (4) Real and repeated roots – i.e. r 1 = r 2 R . Then general solution is ! Note the t followed by c 2 ! y = c 1 e r 1 t + c 2 te r 1 t . (5) Pair of complex conjugate numbers – i.e. r 1 = λ + , and r 2 = ¯ r 1 = λ . Then the general solution is y = e λ t ( c 1 cos μt + c 2 sin μt ) . (6) 1 c hf, 2011
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GE 207K 2nd Order ODEs with Constant Coe ff s September 13, 2011 Examples solved in class: Example 1 Find the general solution to the following ODE: y y = 0 . (7) Solution : Step 1: The corresponding characteristic equation is r 2 1 = 0 . The roots of this quadratic equation are r 1 = +1 , r 2 = 1 Step 2: We have a pair of real and distinct (not repeated roots). Therefore, the general solution is given by Eq. ( 4 ).
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