Problems 3-2 1. A body of weight 32 lb is dropped from a height of 100 ft in a medium offering an air resistance proportional to the velocity. If the limiting velocity is 400 ft/sec, find the velocity and displacement at any time. Find the time at which t
ELEMENTARY DIFFERENTIAL
EQUATIONS
William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus
Department of Mathematics
Trinity University
San Antonio, Texas, USA
wtrench@trinity.edu
This book has been judged to meet the evaluation criteria set by
MAP 2302 Differential Equations I Supplementary Homework Problems 2-2 Separation of variables Solve the following differential equations by separation of variables: 1. dy = cos 3x dx dy = -x + 1 dx
2. dy + e2 ydx = 0 3. (x - 1)
4. x2 y = -2y 5. 6. dx x2 y
Supplementary problems for chapter 2 1. 2. 3. 4. 5. 6. 7.
4 x 6 xy dx 6 x y 3 y dy 0 1 u d 1 du 1 x y dx x y x dy 0
3 2 2 2 2 2 2 2 3
dy y tan x 2 x dx
2 x3 dy 1 3x 2 y dx 0
y 1 e x e y
x 2 y xy 2 y 2 0 dN 8. DE: kN c N dt dN IC: kN c N dt dx 9. DE: a x
Problem 2-7 Solve the differential equations in Probs. 1 through 5. 1.
y 3 y 2
2
2. DE: y 4 y IC: y 0 2 3. 4.
y 0 2
y a 2 y 0
d n 1 y 2 d n y 5x2 n 1 n dx x dx
n a positive integer
5. DE: mx kx mg
IC: x 0 x0
m0 k 0 g 0
0 0
6. Given a general second-orde
Problem 2-6 Solve the following differential equations: 1. 2. 3. 4. 5. 6.
x x
2
2 y 2 dx 2 xydy 0 y 2 dx 2 xydy
2
x y dy x y dx 0
dy 1 e y dx x dy x y xy 4 dx dy y dx x xy
7. By means of an appropriate substitution, reduce the following differential equ
Problem 2-5 Solve by finding an integrating factor: 1. 2. 3. 4.
ydx y 2 x dy 0 xdx y x 2 y 2 dy 0
2 x 3x y dy y x y dx 0
3 2 2
3x y dx 2 xdy 0
2 5. The equation x y dx f x dy 0 is known to have an integrating factor
I x x . Find all possibilities for f
Problem 2-4 1. Test the following for exactness and solve:
2 2 a. x y dy y x dx 0
b. e cos ydx e sin ydy
x x
c. ax by dx bx ay dy 0
2
d.
4 y 2 2 x2 8 y 2 x2 dx 3 2 dy 0 4 xy 2 x3 4y x y
2. Solve: DE: 2 x 1 y 3 dx 3 x 1 y 2 2 y dy 0
IC: y 1 3 3. Derive the
Problem 2-3 1. Find the general solution of: a. y y x c. b. y
y x2 x 1
ds 3s 5 dt t dx e. 2 y x y y3 dy
d. y y tan x sin 2 x f. y y sin x kx
2. Solve: a.DE: y y e x IC: y 1 3 b.DE: y y sin x kx IC: y 0 0
3. A certain first-order linear differential equat
Problem 2-2 1. Solve the following differential equations. The solutions should be put into as simple a form as possible and checked by substitution into the differential equations. It is not necessary(and it is not forbidden) to find the interval for whi
Problems 1-6 1. Solve for : a. DE: IC: b. DE: IC: 2. Find the continuous function except for . DE: IC: 3. Verify that the solution of DE: IC: is
which satisfies the following initial-value problem
4. Derive the solution DE: IC: assuming 5. Show that
of th
Problems 1-5
1. Solve: DE : y x IC:
y 0 0 y 0 1 y 0 0
2. Solve: DE: y 4 y 0 IC: y 0 1
y 0 1
Assume that the general solution of the differential equation is y c1e 2 x c2 e 2 x . 3. The general solution of the differential equation
2 x 0 x
0
is x c1 sin
Problems 1-4 1. Verify that is a solution of
for all
and for all constant
and
.
2. By integrating
times, find the general solution of
3. For what values of does determine as a differentiable function of x which satisfies ? 4. Find the values of , if any,
Problems 1-2
1. Classify the following differential equations as to order, degree, and linearity. State the independent variables and unknown functions.
a. y P x y Q x
b.4 y 3 y 6 y 0
d . y x3
u u f . 2 2 0 x y
d 3s c. 3 5s 4t 3 0 dt
di y d0y e. ai x