Math 224 (Elementary Linear algebra and Differential Equations)
Review problems for Midterm 1
1. Determine the order of
d 2 y x y dy
+e
= x 2 and state whether it is linear or nonlinear.
2
dx
dx
x2 + y2
where x > 0 is a homogeneous function of degree zero or not. (Use the definition}
x− y
2. Determine if
f ( x, y ) =
3. Determine if
[cos( xy ) − xy sin( xy )] dx − x 2 sin( xy ) dy = 0 is exact or not. (Do Not Solve)
4. Solve
dy 2 x ( y − 1)
=
dx
x2 + 3
5. Solve
dy
2x
+
y = 4 x , where −1 < x < 1
dx (1 − x 2 )
6. Solve
dx
2
x = 5 , subject to x(0) = 4
+
dt (4 − t )
7. Solve
x2
8. Solve
( y 2 + cos x) dx + (2 xy + sin y ) dy = 0
dy
= y 2 + 3xy + x 2
dx
0
9. At 2 p.m. (t = 0) on a cool ( 34 F) afternoon in March, Sherlock Holmes measured the temperature of a dead body to
38 F. One hour later the temperature was 360 F. After a quick calculation using Newton’s law of cooling, and
0
taking the normal temperature of a living body to be 98 F, Holmes concluded that the time of death was 10 a.m.
dT
Was Holmes right? [Hint:
= −k (T − Tm ) ]
dt
be
0
1
10. A tank contains 1000 liters (L) of a solution consisting of 100 kg of salt dissolved in water. Pure water is pumped into
the tank at the rate of 5 L /s, and the mixture, kept uniform by stirring, is pumped out at the same rate. Set up (do not
solve) a differential equation which describes the change in the amount of salt in the tank at any time t.
[Hint:
dV
dA
= rin − rout and
= c1 rin − c2 rout ]
dt
dt
1
Farads, and E (t) = 100 volts. If the capacitor is uncharged
50
dq
initially ( q (0) = 0 ), determine the current i(t) in the circuit for t ≥ 0 . [Hint: i(t) =
]
dt
11. Consider the RC circuit which has R = 5 ohms, C =
12. For the first order autonomous differential equation
y ' = y 2 − 7 y + 12
a) Sketch the differential equation in the y − y ' plane.
Draw the phase line diagram, clearly stating the location and stability of the equilibria.
b) State the long term behavior for all initial conditions.
13. By hand, carry out one step of Euler's method for the differential equation
with initial condition
y (−1) = 2 and a step size of 0.1.
dy
= x y − 3y
dx
4. Without solving the equations, match each differential equation with the graph of its direction field shown in either
Figures 1 through 4. (Next page)
a)
dy
= x2 − y2
dx
b)
2
dy
= e0.01x y
dx
c)
dy
= 1− x y
dx
d)
dy
= (sin x) cos y
dx
2
2
2
1
1
0
0
-1
-1
-2
-2
-3
-2
-1
0
1
2
-2
3
-1
0
1
Figure 1
2
Figure 2
3
2
2
1
1
0
0
-1
-1
-2
-3
-2
-3
-2
-1
0
Figure 3
1
2
3
-2
-1
0
1
2
Figure 4
3