MATH 3350-012
Make-up Exam (D-M)
23 February 2010
Name _________________________
Score ___________________
Answer the problems on separate paper. You do not need to rewrite the problem statements on your answer sheets. Work
carefully. Do your own work. Show all relevant supporting steps! Attach this sheet to the front of your answers.
Directions: If the technique you apply to solve a specific problem is to treat the problem as:
a.
b.
1. (10 pts)
a.
a linear differential equation, then specifically identify the component pieces you construct as you solve the
equation, i.e., specifically, the integrating factor
an exact equation, then specifically verify that the equation is exact before proceeding
Classify each of the following differential equations by identifying their order and by identifying whether they
are linear.
x − 2 x=x
d.
d4y d2y
+ 2 + y = ty 2
4
dt
dt
2
3
d y
⎛ dy ⎞
x 2 3 − ( x + 1) ⎜ ⎟ + 6 y = x
dx
⎝ dx ⎠
2
x y ′′′ − 2 xy ′′ + y sin ( x ) = 0
e.
x 2 u ′′ − u ′ + x 2u = ln( x + u )
b.
c.
2. (10 pts)
Find the general solution of the differential equation y ′ + 2 xy + x = 0
3. (10 pts)
Find the general solution of the differential equation
4. (10 pts)
Find the general solution of the differential equation
5. (10 pts)
Find the general solution of the differential equation ( y − yx ) dx + 3 x dy = 0
6. (10 pts)
Solve the initial-value problem
7. (10 pts)
Solve the initial-value problem xy ′ + y = x sin(3 x ), y (1) = 2
8. (10 pts)
Solve the initial-value problem (4 y + 3 x 2 − 6 xy ) dx + (6 y 2 + 4 x − 3 x 2 ) dy = 0, y ( −1) = 2
9. (10 pts)
Find the general solution of the differential equation
2
dy
= 4x + 5 − 3y
dx
y ⎞
1
⎛
2
⎜ x + 3 y − 2 ⎟ dx = (2 y − 3 x − ) dy
x ⎠
x
⎝
2
( x + 1)
2
dy
= 2 xy − xy 2 , y (0) = −1
dx
dy
= 3 y + 2 y3
dx