191_pdfsam_math 54 differential equation solutions odd

# 191_pdfsam_math 54 - 2 = − 1 ± √ 3 i Hence the roots of the auxiliary equation are r 1 = r 2 = − 1 − √ 3 i and r 3 = r 4 = − 1 √ 3 i

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Exercises 4.3 35. The equation of the motion of a swinging door is similar to that for mass-spring model (with the mass m replaced by the moment of inertia I and the displacement y ( t ) replaced by the angle θ that the door is open). So, from the discussion following Example 3 we conclude that the door will not continually swing back and forth (that is, the solution θ ( t ) will not oscillate) if b 4 Ik =2 Ik . 37. (a) The auxiliary equation for this problem is r 4 +2 r 2 +1=( r 2 +1) 2 =0 . Th isequat ion has the roots r 1 = r 2 = i , r 3 = r 4 = i .Thu s ,co s t and sin t are solutions and, since the roots are repeated, we get two more solutions by multiplying cos t and sin t by t ,thatis , t cos t and t sin t are also solutions. This gives a general solution y ( t )= c 1 cos t + c 2 sin t + c 3 t cos t + c 4 t sin t. (b) The auxiliary equation in this problem is r 4 +4 r 3 +12 r 2 +16 r +16=( r 2 +2 r +4) 2 =0 . The roots of the quadratic equation r 2 +2 r +4=0are r =
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Unformatted text preview: 2 = − 1 ± √ 3 i. Hence the roots of the auxiliary equation are r 1 = r 2 = − 1 − √ 3 i and r 3 = r 4 = − 1+ √ 3 i . Thus two linearly independent solutions are e − t cos( √ 3 t ) and e − t sin( √ 3 t ), and we get two more linearly independent by multiplying them by t . This gives a general solution of the form y ( t ) = ( c 1 + c 2 t ) e − t cos( √ 3 t ) + ( c 3 + c 4 t ) e − t sin( √ 3 t ) . 39. (a) Comparing given equation with the Cauchy-Euler equation (21) in general form, we conclude that a = 3, b = 11, and c = − 3. Thus, the substitution x = e t leads to the equation (22) in Problem 38 with these values of parameters. That is, a d 2 y dt 2 + ( b − a ) dy dt + cy = 0 ⇒ 3 d 2 y dt 2 + 8 dy dt − 3 y = 0 . 187...
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## This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at University of California, Berkeley.

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