Differential Equations

1. Show that {e
x
, xex
, e-x} is an independent set of functions on (-8,8).
2. Determine if the following set of functions is linearly dependent or independent on
(-8,8). If it is linearly dependent, then find constants c1, c2, c3, c4, not all 0, such that
c1f1 + c2f2 + c3f3 + c4f4 = 0 on (-8,8).
f1(x) = 5x, f2(x) = 2x + 3, f3(x) = 2x
2 + 3, f4(x) = x
2 + x.
3. Show that the following set of functions is linearly independent on (-8,8).
f1(x) = 1 f2(x) = x f3(x) = x
2
f4(x) = x
3
.
4. Given y
00 + y = 0.
(1) verify that y1 = sin x + cos x and y2 = 4 sin x – 2 cos x are both solutions.
(2) show that y1, y2 are linearly independent.
(3) Find constants c1, c2 so that y3 = cos x can be written as c1y1 + c2y2.
5. Verify that y1 = cos(ln x), y2 = sin(ln x) are linearly independent solution of x
2
y
00 + xy0 +
y = 0 on (0, 8).
6. Below is a second order homogeneous linear differential equation together with one particular
solution y1. Find a second solution y2 that is linearly independent of y1. Then write
down the general solution of the D.E.
2x
2
y
00 + 3xy0 – y = 0, y1 = x
1/2
.
7. Below is a second order homogeneous linear differential equation together with one particular
solution y1. Find a second solution y2 that is linearly independent of y1. Then write
down the general solution of the D.E.
xy00 + (2x – 1)y
0 – 2y = 0, (x > 0), y1 = e
-2x
.
8. y
00 – 3y
0 + 2y = sin x
9. y
00 – 3y
0 + 2y = xe2x
10. y
00 – 3y
0 + 2y = 2x
2 + 3e
2x
11. y
00 + 4y
0 + 4y = 3e
-2x + 4x
2
12. Write down just the form of yp for the following equation. Don’t solve for the constantsor
solve the equation.
y
00 – 3y
0 + 2y = x
2
e
x + e
2x
(II) Solve the following differential equations using variation of parameters.
13. y
00 + y = sec2 x
14. y
00 – 2y
0 + y =
e
x
1+x2
.
15. y
00 – 9y =
9x
e
3x .
16. A mass weighing 6 pounds stretches a spring 2 feet. The mass is initially released from
a point 2 foot below the equilibrium position with a downward velocity of 8 ft/s.
(1) Find the equation of motion. Write the displacement function x = x(t) in the form of
A sin(?t + f). Find the amplitude A, period, and the phase angle f.
(2) When is the first time the mass returns to the equilibrium position, with what velocity?
(3) When is the maximum displacement from the equilibrium position first reached?