is a nonzero solution to , obtain a second, linearly independent solution by reduction of order. (b) Use the Wronskian to prove the two solutions are linearly independent. (a) Set y = vx . Substituting y into the ODE above and simplifying the algebra results in Then, setting w = v ′ , and a little additional algebra provides us with the linear, 1st order, homogeneous equation An integrating factor for this ODE is μ = x 2 /( x 2- 1). Solving the ODE, an additional integration, and a suitable choice of constants allows us to obtain a second solution Hence, f and g are linearly independent functions. ______________________________________________________________________ 9. (10 pts.) An 8-lb weight is attached to the lower end of a coil spring suspended from the ceiling and comes to rest in its equilibrium position, thereby stretching the spring 0.4 ft. The weight is then pulled down 6 inches below its equilibrium position and released at t = 0. The resistance of the medium in pounds is numerically equal to 2x ′ , where x ′ is the instantaneous velocity in feet per second. (a) Set up the differential equation for the motion and list the initial conditions. (b) Solve the initial-value problem set up in part (a) to determine the displacement of the weight as a function of time. (a) Let x ( t ) denote the displacement, in feet, from the equilibrium position at time t , in seconds. Since the mass, m = 8/32 = 1/4 slug, and the spring constant is k = 20 pounds per foot, an initial-value problem describing the motion of the mass is (b) The ODE is equivalent to x ″ + 8x ′ + 80x = 0 with as the general solution. Using the initial conditions to build a linear system and solving the system provides us with the solution to the IVP, . ___________________________________________________________________________ Bonkers 10 Point Bonus: Obtain a condition that implies that will be an integrating factor of the differential equation and show how to compute μ when that sufficient condition is true. Say where your work is for it won’t fit here.
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