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midterm_2_review (dragged) 3 - 4 MA 36600 MIDTERM#2 REVIEW...

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4 MA 36600 MIDTERM #2 REVIEW § 3.6: Variation of Parameters. To find the general solution of the nonhomogeneous equation a ( t ) y + b ( t ) y + c ( t ) y = f ( t ), perform the following steps: #1. Find 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 ) d τ + c 1 and u 2 ( t ) = t f ( τ ) a ( τ ) y 1 ( τ ) W ( τ ) d τ + c 2 in terms of W ( t ) = y 1 ( t ) y 2 ( t ) y 1 ( t ) y 2 ( t ). #3. Form 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 ) d τ . § 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 = m g . Restoring Force: Hooke’s Law states that F s = k ( L + u ) for some positive constant k .
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