midterm_2_review (dragged) 3

Midterm_2_review - 4 MA 36600 MIDTERM#2 REVIEW 3.6 Variation of Parameters To nd the general solution of the nonhomogeneous equation a(t y b(t y

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4M A 3 6 6 0 0 M I D T E R M # 2 R E V I E W § 3.6: Variation of Parameters. To fnd the general solution oF the nonhomogeneous equation a ( t ) y °° + b ( t ) y ° + c ( t ) y = f ( t ), perForm the Following steps: #1. ±ind a Fundamental set oF solutions { y 1 ,y 2 } to a ( t ) y °° + b ( t ) y ° + c ( t ) y = 0. #2. Compute the integrals u 1 ( t )= ° t f ( τ ) a ( τ ) y 2 ( τ ) W ( τ ) + c 1 and u 2 ( t )= ° t f ( τ ) a ( τ ) y 1 ( τ ) W ( τ ) + c 2 in terms oF W ( t )= y 1 ( t ) y ° 2 ( t ) y ° 1 ( t ) y 2 ( t ). #3. ±orm the Function y ( t )= u 1 ( t ) y 1 ( t )+ u 2 ( t ) y 2 ( t ). This method is called Variation of Parameters . We can always write y ( t )= u 1 ( t ) y 1 ( t )+ u 2 ( t ) y 2 ( t )= c 1 y 1 ( t )+ c 2 y 2 ( t )+ Y ( t ) in terms oF the particular solution Y ( t )= ° t f ( τ ) a ( τ ) y 1 ( τ ) y 2 ( t ) y 1 ( t ) y 2 ( τ ) y 1 ( τ ) y ° 2 ( τ ) y ° 1 ( τ ) y 2 ( τ ) dτ. § 3.7: Mechanical and Electrical Vibrations. Say that we have a mass m which is attached to a spring with constant k . Consider Four Forces on the mass: Gravity: Newton’s Law oF Gravity states that F g = mg .
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.

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