PRACTICE TEST 3 Math 336 1. Assume that a mass of 1 slug is attached to a spring with a constant 9 lb/ft and that no damping force acts on the system. (a) If the mass is driven by external force f(t) = sin(wt), where w > 0 is the frequency of the force, explain why the equation of motion is x"+9x=sin(wt). (b) If w3 find the general solution of the equation from (a). (c) Find the value of w for which resonance occurs. Answer: x(t) = C₁ cos(3t) + C2 sin(3t) +92 sin(wt) 2. Find the Laplace transform of the following functions using either the table or the definition as specified. (a) Find the Laplace transform of the function u(t - 10) using the definition. (b) f(t) = e(t + 5 sin(2t)) Answer: F(s) = (8-3)² + (c) f(t)=t2tet Answer: F(s) = (-1)² 10 (8-3)2+4 3. Find the inverse Laplace transform of the following functions: (a) F(s) = 8+5 s2+28 +5 Answer: f(t) e cos(2t) + 2et sin(2t) (b) F(s) = 1 s(s²+4) Answer: f(t) (1 - cos(2t)) = 4. Compute the convolution of f(t) = est and g(t) e-2 using the definition. Answer: f(t)g(t) = (et - e−2). 5. Use the Laplace transform to solve the initial value problem: x" +4x=e-2t where the initial condition is x(0) = 1, x'(0) = -1. Answer: x(t)=(e-2 +7 cos 2t - 3 sin 2t) 6. Use the Laplace transform to solve the initial value problem: "+4x+8x= u(t-2) where the initial condition is x(0) = 0, x'(0) = 0. Answer: x(t)=(-2) [1 - e-2(-2) (cos(2(t − 2)) + sin(2(t − 2)))] 1