# SampleQ_2p - t ): ( D 2 + 4 . 3 D + 31 I ) θ = f ( t )...

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MATH 405: Sample Midterm Questions 1. Show that the following two functions are linearly independent (hint: use the Wron- skian) x 1 . 3 x 1 . 3 log x 2. Find the homogeneous and particular solutions for the following equation for y ( t ) y 00 + 2 y 0 + 31 y = e - t sin t What is the long term (steady state) behavior of y ? 3. Show that the second order ODE for an LRC circuit: LI 00 + RI 0 + 1 C I = 0 can be written as two coupled ﬁrst order ODEs. 4. Consider the set S of all polynomials of degree 8 (powers of x 8 and lower). Is this a vector space? What is the dimension of S ? Give a basis. 5. Consider the following matrix B = 0 -2 8 2 0 -1 -8 1 1 Find the rank of B and a basis for the column space. 6. Solve the ﬁrst order system of diﬀerential equations y 0 = Ay for A = ± 1 -12 4 1 ² 7. Consider the oscillations of a damped linear pendulum described by θ (

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Unformatted text preview: t ): ( D 2 + 4 . 3 D + 31 I ) θ = f ( t ) where f is an additional forcing function. In addition there are initial conditions θ (0) and θ (0). (a) If f ( t ) = A sin ωt , what numerical value of ω would be the resonant frequency? (b) If you wanted the pendulum to more quickly adjust itself to another driving fre-quency ω 1 (not the resonant one), what would you change in the equation? 1 8. Given the matrix: A = ± i 4 i 1 + 4 i ² decompose it as A = H + X , where H is a Hermitian matrix and X is skew-Hermitian. 9. Find a general solution of x 2 y 00-3 y = 0 10. Can the following matrices be diagonalized? Answer without diagonalizing! Q = ± 3 2 3 6 ² R = ± 5 1 1 5 ² T = ± 5 1 5 1 ² 2...
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## This note was uploaded on 07/23/2008 for the course MATH 405 taught by Professor Belmonte during the Fall '06 term at Pennsylvania State University, University Park.

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SampleQ_2p - t ): ( D 2 + 4 . 3 D + 31 I ) θ = f ( t )...

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