midterm_2_review (dragged) 2

midterm_2_review (dragged) 2 - MA 36600 MIDTERM #2 REVIEW 3...

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MA 36600 MIDTERM #2 REVIEW 3 A fundamental set of solutions to the diFerential equation is { y 1 ,y 2 } in terms of the functions y 1 ( t )= ° e λt cosh μt if b 2 4 ac> 0, e λt cos μt if b 2 4 ac< 0. y 2 ( t )= ° e λt sinh μt if b 2 4 ac> 0, e λt sin μt if b 2 4 ac< 0. § 3.4: Repeated Roots; Reduction of Order. Consider the constant coefficient diFerential equation ay °° + by ° + cy =0 . Say that the quadratic polynomial ar 2 + br + c has discriminant b 2 4 ac = 0 i.e., the characteristic equation has repeated roots λ = b/ (2 a ) . We know that one solution is y 1 ( t )= e λt , so we may guess that all solutions are in the form y ( t )= v ( t ) y 1 ( t ) for some function v = v ( t ); this is d’Alembert’s Method of Reduction of Order . We ±nd that v °° =0i.e., v ( t )= c 1 + c 2 t . In particular, the real-valued functions y 1 ( t )= e λt and y 2 ( t )= te λt form a fundamental set of solutions. § 3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients.
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