Page 23 - Schaum's Outline of Differential Equations

6 BASIC CONCEPTS [CHAR 1

Since

To satisfy the condition we require ar equivalently,

Solving (_/) and (2) simultaneously, we obtain and

1.13. Determine c 1 and c 2 so that will satisfy the conditions y(0) = 0 and /(O) = 1.
Because sin 0 = 0, y(0) = Cj + c 2. To satisfy the condition y(0) = 0, we require

From

we have / (0) = 2cj + c 2 + 2. To satisfy the condition / (0) = 1, we require 2cj + c 2 + 2 = 1, or

Solving (_/) and (2) simultaneously, we obtain and

Supplementary Problems

In Problems 1.14 through 1.23, determine (a) the order, (b) the unknown function, and (c) the independent
variable for each of the given differential equations.

1.24. Which of the following functions are solutions of the differential equation y' — 5y = 0?
5x
5x
5
(a) y = 5, (b) y = 5x, (c) y = x , (d) y = e , (e) y = 2e , (/) y = 5e 2x
1.25. Which of the following functions are solutions of the differential equation y' — 3y = 6?
3x
2x
3x
(a) y = -2, (b) y = 0, (c) y = e -2, (d) y = e -3, (e) y = 4e -2