Question: k(t) = 6 -1 N/m m 2 kg 1 N-sec/m x(1) x(0) = 3 m...

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Question: k(t) = 6 -1 N/m m 2 kg 1 N-sec/m x(1) x(0) = 3 m x'(0) = 0 m/sec Figure 1: A mass-spring system whose spring is being heated Variable Spring Constant. As a spring is heated, its spring "constant" decreases. Suppose the spring is heated so that the spring "constant" at time t is k(t) = 6 - t N/m (see Figure 1). If the forced mass-spring system has mass m = 2 kg and a damp- ing constant b = 1 N-sec/m with initial conditions x(0) = 3 m and x' (0) = 0 m/sec, then the displace- ment x(t) is governed by the initial value problem 2x" (1) + x' (t) + (6 - t)x(t) = 4t x(0) = 3, x'(0) = 0. Find at least the first four nonzero terms in a power series expansion about t=0 for the displacement. Hence, calculate the displacement x(t) when time t is 2 seconds. [Estimate x(2) using the first 4 terms of x(t) only] Question: k(t) = 6 -1 N/m m 2 kg 1 N-sec/m x(1) x(0) = 3 m x'(0) = 0 m/sec Figure 1: A mass-spring system whose spring is being heated Variable Spring Constant. As a spring is heated, its spring "constant" decreases. Suppose the spring is heated so that the spring "constant" at time t is k(t) = 6 - t N/m (see Figure 1). If the forced mass-spring system has mass m = 2 kg and a damp- ing constant b = 1 N-sec/m with initial conditions x(0) = 3 m and x' (0) = 0 m/sec, then the displace- ment x(t) is governed by the initial value problem 2x" (1) + x' (t) + (6 - t)x(t) = 4t x(0) = 3, x'(0) = 0. Find at least the first four nonzero terms in a power series expansion about t=0 for the displacement. Hence, calculate the displacement x(t) when time t is 2 seconds. [Estimate x(2) using the first 4 terms of x(t) only]