de-fx-a

# Is a nonzero solution to obtain a second linearly

This preview shows page 6. Sign up to view the full content.

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.
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Ask a homework question - tutors are online