Math 34 Differential Equations Quiz 3 SHOW ALL WORK

1.) Define: The LaPlace transform of f = L(f) = \underline{∫_{0}^{∞} e^{−st}f(t)dt}

2a.) L(0) = \underline{0}

2b.) L^{−1}(\frac{3}{(s+1)^{2}+2}) = \underline{\frac{3e^{−t}}{\sqrt{2}} sin(\sqrt{2}t)}

L^{−1}(\frac{3}{(s+1)^{2}+2}) = 3L^{−1}(\frac{1}{(s+1)^{2}+2}) = \frac{3}{\sqrt{2}}L^{−1}(\frac{\sqrt{2}}{(s+1)^{2}+2}) = \frac{3e^{−t}}{\sqrt{2}} L^{−1}(\frac{\sqrt{2}}{(s)^{2}+2}) = \frac{3e^{−t}}{\sqrt{2}} sin(\sqrt{2}t)

3.) Circle T for True or F for False: 3a.) Suppose y = f(t) is a solution to y′′ + y′ + y = cos(2t), y(0) = 0, y′(0) = 0, and suppose y = g(t) is a solution to y′′ + y′ + y = cos(2t), y(0) = 100, y′(0) = −200. For large values of t, f(t) − g(t) is very small. T 3b.) The initial conditions have a transient effect on the solution to y′′ + y = cos(2t).

F

3c.) The initial conditions have a transient effect on the solution to y′′ −y′ +y = cos(2t). F

4.) Match the following differential equation initial value problem to its graph: B = I.) y′′ + y = cos(t), y(0) = 0, y′(0) = 0 A = II.) y′′ + y = cos(1.2t), y(0) = 0, y′(0) = 0

II = A.) I = B.)