Math 215
HW1
March 25, 2016
Note: All derivatives, y 0 , y 00 , u0 , u00 , are with respect to x.
(1) Suppose that y is a function of x. Express the following in terms of x, y 0 and y 00 .
d 2
(a)
(y )
dx
d2 2
(y )
(b)
dx2
d 2
(c)
(x y)
dx
(2) Suppose tha
Math 215
HW4
April 9, 2016
(1) Solve the following differential equations. All are exactperhaps after a bit of algebraic manipulation.
dy
(a) (2xy 2 + 2y) + (2x2 y + 2x)
=0
dx
y
+ 6x dx + (ln x 2) dy = 0 with x > 0.
(b)
x
dy
ax + by
(c)
=
with a, b, c, d
Math 215
HW12
May 16, 2016
(1) Find particular solutions
of the following:
(a) (D 1)2 y = ex x.
(b) (D 1)y = ex ln x.
(c) (D 1)y = 2 sin2 x.
(2) Let u and v be functions of x that satisfy u0 = xv and v 0 = xu.
(a) Find second order linear differential equ
Math 215
HW5
April 13, 2016
In a leaking water tank, the height of water h is a decreasing function
of time t. Using Torricellis Law, we derived in class the differential
equation satisfied by h:
p
dh
A
= a 2gh
dt
where g is the acceleration of gravity (9
Math 215
HW7
April 21, 2016
(1) Solve the following differential equations.
(a) y (4) y = 0
(b) y (4) + y 00 = 0
(c) y (4) y 00 = 0
(d) y (4) + y 000 = 0
(e) (D2 2)y = D2 y
(f) (D3 5D2 + 9D 5)y = 0
d2 y
dy
(g)
2
+ 5y = 0
2
dx
dx
2
(h) (D 4D + 20) y = 0 wi
Math 215
HW9
May 5, 2016
(1) Find the general solutions of the following differential equations.
(a) y 00 + y 0 2y = e2x .
(b) y 00 + y 0 2y = 2x.
(c) y 00 + y 0 2y = e2x + 2x.
(d) y 00 + 2y 0 = 3 + 4 sin 2x
(e) (D2 + 1)y = x2 + x + 1
(f) (D 1)y = sin2 x.
Math 215
HW3
April 6, 2016
(1) Solve the following differential equations. Some are separable, some are linear, some are
both. If possible, express y as a function of x.
dy
(a) x3 3y x
=0
x>0
dx
(b) y + x
dy
= 2e2x
dx
(c) 9x2 y 2 + x3/2
dy
= y2
dx
x>0
(d)
Math 215
HW8
April 26, 2016
d2 Q
dQ
1
In class we discussed the differential equation L 2 + R
+ Q = 0. We set = R/2L and
dt
dt
C
d2 Q
dQ
2
+ 2
0 = 1/ LC which simplified the equation to
+ 0 Q = 0, and then we found the
dt2
dt
general solution in the case
Math 215
HW10
May 17, 2016
d2 Q
(1) In class we discussed the differential equation
+ 02 Q = a cos t and found the
dt2
general solution in the case that 6= 0 . What is the general solution when = 0 ?
(2) Find particular solutions of the following differen
Math 215
HW14
May 24, 2016
(1) Using the variation of parameters method, find particular solutions of the following:
(a) y 00 4y = e3x . Note: You solved this already in HW11(1a) using operator methods
(which is easy). Now try the variation of parameters
Math 215
HW13
May 19, 2016
(1) Using the reduction of order method, find the general solution of the following differential
equations given the particular solution y1 .
(a) (D 3)2 y = 0, y1 = e3x .
(b) x2 y 00 + 2xy 0 = 0, y1 = 1.
(c) xy 00 y 0 x3 y = 0,
Math 215
HW11
May 12, 2016
(1) Find particular solutions of the following differential equations using the operator method.
(a) y 00 4y = e3x
(b) y (4) = e3x
(c) y 00 4y = e2x
(d) y 0 2y = xe2x
(e) y 00 4y 0 + 4y = xe2x
(f) (D 1)y = ex cos x.
(2) Find par
Math 215
HW2
April 4, 2016
(1) Verify thatthe following functions
are solutions of the corresponding differential equations.
(a) y = 2 x + c, y 0 = 1/ x.
(b) y 2 = e2x + c, yy 0 = e2x .
p
d
(c) y = arcsin xy, xy 0 + y = y 0 1 x2 y 2 . Hint: arcsin x = sin