Question

DIFFERENTIAL EQUATIONS

1. A force of 400 newtons stretches a spring 2 meters. A mass of 50 kilograms is attached to the end of the
spring and is initially released from the equilibrium position with an upward velocity of 10 m/s. Find the equation of
motion.

2. A 4-foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through
which the mass moves offers a damping force numerically equal to times the instantaneous velocity. Find the
equation of motion if a mass of ¼ slug is initially released from the equilibrium position with a downward
velocity of 5 ft/s.

3. A mass weighing 16 pounds stretches a spring feet. The mass is initially released from rest from a point 2 feet
below the equilibrium position, and the subsequent motion takes place in a medium that offers a damping force
that is numerically equal to 1/2 the instantaneous velocity.
The mass is driven by an external force equal to F(t) = 10 cos 3t
(a) The complementary solution is: _________________________________________
(b) The particular solution is: _______________________________________________
(c) The general solution is: ___________________________________________________

4. A beam is free on left and embedded on right, and wx=w0, 0<x<L, and governed by EId4ydx4=wx

Find y(x).

Hint: use boundary conditions in this order:

            Boundary condition:

            Boundary condition:

            Boundary condition:

            Boundary condition:

5. Find the first 4 terms of the two power series solutions of the DE about the ordinary point x = 0.

y''-2xy'+y=0
You can write the coefficients of your two answers as numbers rather than in terms of Co and C1

That is y(x) = 1- 1/4x^2 + ..... instead of y(x) = Co [ 1- 1/4x^2 + ....]

6. Find the first 3 terms of the two power series solutions to the IVP about the ordinary point x = 0.

You can write the coefficients of your two answers as numbers rather than in terms of

(x-1)y'' -xy' + y = 0 , y(0) = -2, y' (0) = 6

7. Find the indicial roots of 3.
Roots: ___, _____
What do we expect the two series solutions to look like? Fill in the blanks.

y = ____ * ( Co + C1x + C2x + ........ ) + _______*(bo + b1x +b2x +.....)

Answer 1