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# midterm2 - 18.034 Midterm#2 Name Part I TF Questions Answer...

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18.034 Midterm #2 Name: Part I: TF Questions. Answer for each of the following statements if it is true or fale. Simply say T (if you believe it is true) or F (if you suspect it is false); you don’t need to justify your answers. Each question counts 5pts. 1. Let the differential operator L [ y ] = y ( n ) + a 1 y ( n 1) + + a n y where a 1 , . . . , a n are real numbers, · · · have the characteristic polynomial p ( λ ) = λ n + a 1 λ n 1 + · · · a n . If p ( α ) = 0 then L [ y ] = e αx has a solution of the form Ae αx . 2. The solution of the initial value problem y ��� + y �� 2 y = δ ( t ) with y (0) = y (0) = y �� (0) = 0 decays exponentially as t → ∞ . Here, δ ( t ) denotes the Dirac delta function (or the unit impulse function). 3. The curves defined parametrically as solutions of the system dx y dy 1 dt = x 2 , dt = x are orthogonal to the surface y/x = constant. 4. The function f ( x ) = x log x 0 0 < x 1 , x = 0 satisfies a Lipschitz condition. 2 5. The solution of the initial value problem x = t 2 + x , x (0) = 0 exists for | t | ≤ 1 / 2 .

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midterm2 - 18.034 Midterm#2 Name Part I TF Questions Answer...

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