Section 7.6

# Section 7.6 - Differential Equations Math 308 Spring 2008...

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Unformatted text preview: Differential Equations Math 308, Spring 2008 Texas A&M University c F. Dos Reis Section 7.6 Definition: The unit step function u(t) is defined by u(t) = 0 t<0 1 t>0 Exercise 1. Describe the function v(t) = M u(t - a) Find the Laplace transform of u(t - a). Theorem: L {u(t - a)} (s) = L-1 e-as s e-as s (t) = u(t - a) Exercise 2. The function g(t) is defined by 2 t<5 6 5 < t < 10 g(t) = 2 t 10 < t Express g using unit step functions. Theorem: If f is of exponential order and a is a positive real number, then L {u(t - a)f (t - a)} (s) = e-as L {f (t)} s, Equivalently, if g is of exponential order L {u(t - a)g(t)} (s) = e-as L {g(t + a)} s for s > for s > Exercise 3. Determine the Laplace transform of g(t) = sin(3t)u(t - ). h(t0 = t2 u(t - 1) 1 Differential Equations Math 308, Spring 2008 Exercise 4. Determine the inverse Laplace transform of F (s) = e-2s F (s) = e-s 3s . s2 +9 Texas A&M University c F. Dos Reis 24 . s4 Exercise 5. Solve the initial value problem y + y = u(t - 3) with y(0) = 0 and y (0) = 1. Theorem: If f has period T and is piecewise continuous on [0, T ], then L {f } (s) = T 0 e-st f (t)dt 1 - e-sT Exercise 6. Let f be a periodic function with period 2 such that f (t) = e3t for t [0, 2). Find the Laplace transform of f . 2 ...
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Section 7.6 - Differential Equations Math 308 Spring 2008...

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