1. Review Problems for the Final Exam (Fall 2014)
1. Solve the equation:
dy
+ (1 − x)2 y = xe−x
dx
−x
−x
−x 2
Solution: y(x) = C(e + e x ) + −e2 .
2. Explain the method for solving linear differential equations of the form
(1 + x2 )
xr y (r) + a1 xr−1 y (r−1) + · · · + ar y = 0,
where ai are constants.
Solution: This is a generalized Euler equation (see Chapter 4.).
3. By using any method solve the equation
y 000 − 4y 0 = cos(x) + ex .
Solution: y = C1 e−2x + C2 e2x + C3 + −1
sin(x) − 13 ex
5
4. By using power series method solve
y 00 − 2xy 0 + 4y = 0, y(0) = 1 y 0 (0) = 1.
5. Find all solutions of
x2 y 00 + xy 0 + 4y = x, x > 0.
by using the variation of constants method.
Solution: y = C1 x2i + C2 x−2i + x5 or y = C1 sin(2ln(x)) + C2 cos(2ln(x)) + x5 .
6. Find a second linearly independent solution of
(1 − x)y 00 − 2xy 0 − 2y = 0, x > 1
1
if φ(x) = x−1
is a solution. Then write the general solution.
−2x
c1
Solution: y = x−1
+ c2 ex−1
7. Show that x = 0 is a regular singular point of
x2 y 00 + (sin(x))y 0 + (cos(x))y = 0,
but irregular for
x2 y 00 + (cos(x))y 0 + (sin(x))y = 0,
3
Solution: Observe that sin(x) = x − x3! + · · · , so sin(x)/x2 = 1/x + · · · , so the first
2
equation is regular at x = 0. But cos(x) = 1 − x2 + · · · , so the second equation is irregular
at x = 0.
8.
(a) Solve
(x − 4)y 4 dx − x3 (y 2 − 3)dy = 0
1
2
(b) Use substitution u = y/x to solve
dy
x2 − xy + y 2
=
dx
xy
(c) Solve
dy
−ey
= y
dx
xe + 2y
9. Solve the initial value problem
− 23 12
y =
y
1 −1
5
y(0) =
.
4
0
Solution:
y=
3e−x/2 + 2e−2x
6e−x/2 − 2e−2x
.
10. By using any method solve
y10 = y2 + x
y20 = 2y1 − y2 + x2
Solution:
y1 (x) = C1 e−2x + C2 ex −
x2
3
− x − , y2 (x) = C2 ex − 2C1 e−2x − 1 − 2x
2
2
11. Solve
y (4) + y = 0
by using real functions (sin and cos).
x
x
√
√
− √x
− √x
Solution: y = c1 e 2 sin( √x2 ) + c2 e 2 cos( √x2 ) + c3 e 2 sin( √x2 ) + c4 c1 e 2 cos( √x2 ).