#### Question Details

Math 331.07 Homework 2: First order ODEs Due Thursday 9/29 Name:

1. Find the order of the following differential equations (y = y(x)). Do NOT solve them:

(a) xy 00 + y = x3

(b) y 0 + y 2 = 2

(c) sin y 000 + 3x2 y = 6x

(d) y 00 (y 0 )1/2 ? 1/y = 0

2. Verify that the function y = y(x) given is a solution of the following ODEs.

(a) y 0 + 4y = 1.4,

(b) y 0 = y + ex , y = e?4x + 0.35

y = (1 + x)ex (c) y 0 tan x = 2y ? 8, y = 4sin2 x + 4 3. Solve the following IVP:

dy

= 4e?t cos t,

dt y(0) = 0 4. Find the general solution to the following Separable ODEs:

(a) y 3 y 0 + x3 = 0

(b) y 0 sin (2?x) = ?y cos (2?x)

5. Solve the following initial value problems (IVPs):

(a) xy 0 = 2 (y ? 4),

0 y(1) = 1 (b) y = 2y (2 ? y), y(0) = 1 (c) y 0 y = 4 sin (2x), y(0) = 1 6. The population of the US was 8.6 Million in 1820 and 40 Million in 1897. Consider

that the rate of change of the population is proportional to the population.

(a) Write down the differential equation for population growth in this case.

(b) Calculate the number of people in the US according to your model in the year

2003. Note that the actual population of the US in 2003 was 291 Million.

7. Boyle-Mariotte?s law for ideal gases (found experimentally by physicists Robert Boyle

and Edme Mariotte) states that for a fixed amount of an ideal gas kept at constant

temperature and low pressure p, the rate of change of the volume V = V (p) equals

?V /p. Solve the model and then explain your outcome. Math 331 Homework Assignment 2 Page 2 of 2 8. Find the general solution to the following first-order linear ODEs. In addition, find

the particular solution whenever an initial condition is provided.

(a) y 0 + y = 1

.

1+ex (b) xy 0 + (x + 1) y = x, with y(ln 2) = 1.

(c) y 0 ? 2y ? x2 e2x = 0, with y(0) = 0.

9. Solve the initial value problem (IVP)





t2 + 4 y 0 + 2ty ? t2 t2 + 4 = 0, y(0) = 3/5.

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