M302 Hw10 (S. Zhang) .
3. (4.3:13)
Find a yp form for a particular solution.
1. (4.3:1)
y 2y + y = t3 + 2et
Find the general solution.
y
y y + y = 2et + 3
ans: Characteristic equation for yc :
r3 2r2 + r = 0
ans: Characteristic equation for yc :
r3 r2
ans: Plug y1 = t2 , y2 = t1 into the equation, and
we can see they are solutions.
M302 Hw8 (S. Zhang) .
1. (3.7:3)
t2 y 2y = 0
Find the general solution. Here you need to nd yp twice,
by both the method of undetermined coecients and the
method of variati
M302 Hw9 (S. Zhang) .
4. (3.9:11)
A srping is stretched 6 in. by a mass that weights 8 lb.
The mass is attached to a dashpot mechanism that has a
damping constant of 0.25 lb-sec/ft and is acted on by an
external force of 4 cos 2t lb.
(a) Determine the ste
M302 Hw2 (S. Zhang) .
gal/min, with the mixture again leaving at the same rate.
Find the amount of salt in the tank at the of an additional
10 min.
1. (2.3:a1)
Suppose that a motorboat is moving at 40ft/s when its
motor suddenly quits and that 10 s later
M302 Hw3 (S. Zhang) .
2y 1/2
1. (2.4:a1)
Determine if the existence and uniqueness theorem for the
rst order DE guarantee the solution (in a neighborhood)
for each of the initial value problems:
dy
(a)
= ln(x y + 1);
dx
dy
= ln(x y + 1);
(b)
dx
dy
(c)
= l
M302 Hw1 (S. Zhang) .
with(DEtools):
DEplot( diff(y(x),x)=y(x)-x,y(x),
x=-5.5,[y(4)=0],y=-5.5,stepsize=.05);
1. First construct a slope eld (3 points are required: (0, 1/2)
, (1, 2) , (1, 0).) Then sketch the solution curve corresponding to the given inti
Note, the above part can be omitted if you can do the second part below correctly. Because the second part, found
a solution, implies the equation is exact!
M302 Hw4 (S. Zhang) .
1. (2.6:a1)
Show the dierential equation is exact, and solve it as an
exact
By the dierential equation
M302 (S. Zhang) .
1.
48.8
0
0
Solve the system x = Ax + g by the eigenfunctions and the undertermined coecients:
A=
=
2
3
1 0
, g=
1 1
A1
A1 A 2
+
2
3
0 = A1 + 2
0 = A 1 A2 + 3
ans:
det(A rI) = 0, r = 1
A1 = 2
For r = 1
1 2
0 0
M302 Hw17 (S. Zhang) .
Find limt x(t).
ans:
1. (7.8:3)
Find the general solution of the homogeneous system by
the method of eigenvalue:
|A rI| = 0
3/2
1
x
1/4 1/2
x =
r = 1/2, 1/2
We solve (A rI)x = 0
Find limt x(t).
5/2 5/2
5/2 5/2
ans:
|A rI| = 0
v1 =
M302 Hw15 (S. Zhang) .
with(linalg):
A:=matrix(2,2,[2,-5,1,-2]);
eigenvectors(A);
dsolve(cfw_diff(x(t),t)=A[1,1]*x(t)+A[1,2]*y(t),
diff(y(t),t)=A[2,1]*x(t)+A[2,2]*y(t),
cfw_x(t),y(t));
1. (7.6:2)
Solve the system x = Ax by eigenfunctions. Find the
limit w
For r = 2, solving (A rI)x = 0 for eigenvectors:
M302 Hw14 (S. Zhang) .
1. (7.3:7)
1 i | 0
i 1 | 0
Find the linear dependence of vectors a1 , a2 (and a3 ) by
the following two methods, where A is formed by the vectors as its columns. (1) det A, (2) nonzer
M302 Study Guide (Final) (S. Zhang) .
By the DE y = only when
1. Solve
6 2y = 0
2
2xy 3y = 9x , y(1) = 3
9 = t2 + 9
t2 = 0
t=0
3
9
y= x
2x
2
The integrating factor is
=e
R
3/(2x)
Since the answer is interval containing t0 = 1, (0, ).
= e(3/2) ln x = x3/2
4. Find a linear homogeneous constant-coecient equation
so that the function is a solution.
M302 Study Guide 2 (S. Zhang) .
1. Find a constant coecient, homogeneous dierential equation so that
y(x) = e2x sin x + 4x2
(1)y = e3t 7e3t
ans: Roots are
(2)y =
M302 Exam 302(February 5, 2008) version 1
You MUST return this sheet with your name. Please write
all solutions in order.
1. (10 pts) Show the linearly dependence (1) by Wronskian, (2) directly by nding a zero linear combination
with nonzero coecients.
4.
M302 Study Guide 1 (S. Zhang) .
3. Carbon extracted from an ancient skull contained only
one-six as much Carbon14 as carbon extracted from
present-day bone. How old is the skull? (The half line
of of Carbon14 is 5700 years.)
1. Solve
2xy 3y = 9x2 , y(1) =
M302 Hw11 (S. Zhang) .
Dierentiate the equation again, and again, we get
1. (5.3:1)
y = (sin x)y 2(cos x)y + (sin x)y
y (0) = 0 2(1) + 0 = 2
Find the rst 4 nonzero terms in the series solutions (without nding the recurrent relation)
y (4) = (sin x)y
y + x
M302 Study Guide 3 (S. Zhang) .
So x = 0 is a regular singular point.
1.
Find the linear dependence (on the whole real line) by
both the Wronskian and the direct solution of coecients.
Let y = xr , then x 0 (we can do it directly or usinig
formula), the i