Differential Equations Exam Review (2 of 6) - Review 1:...

Review 1: L1-L7
1. Determine which of the equations given below (the independent variable is specified)
is
1) linear;
2) separable;
3) both linear and separable;
4) neither linear nor separable.
A.
3 t
dx
x t x e
dt
+ =
B.
2
1
1
dy
x y
x dx
− =
+
C.
0 ydy xdx − =
D.
x
dy
x e
dx
=
(t – independent) (x – independent) (x – independent) (x – independent)
2. Verify whether the function
( )
2 1 x x x φ − = + is an explicit solution to the differential
equation
2
2 2
2 d y y
dx x
= .
3. Determine whether the given relation is an implicit solution to the differential
equation:
3 2
ln x y y − =
2 3
2ln
dy x y
dx y y
=
+
4. Determine whether the Uniqueness and Existence Theorem implies that the given
initial value problem has a unique solution.
4
5. Solve the initial value problem in #4. How many solutions do you have? Give each
solution in explicit form.
6. Determine whether the Uniqueness and Existence Theorem for a linear first-order
differential equation implies that the given initial value problem has a unique solution. If