Page 36 - Schaum's Outline of Differential Equations

CHAP. 3] CLASSIFICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 19

riere M9x,y) = 1/x, N(x,y) = -1/y

and the equation is exact. Thus, a given differential equation has many differential forms, some of which may be
exact.

3.13. Prove that a separable equation is always exact.
For a separable differential equation, and Thus,

and

Since 3M3;y = 3AV3;c, the differential equation is exact.

3.14. A theorem of first-order differential equations states that if f(x, y) and df(x, y)ldy are continuous in a
rectangle 2ft: \x - x 0\ < a, \y — y 0\ < b, then there exists an interval about x 0 in which the initial-value
problem y' =f(x, y); y(x Q) = y Q has a unique solution. The initial-value problem y' = 2^J\ y \',y(0) = 0 has
the two solutions y = x \x\ and y = 0. Does this result violate the theorem?
No. Here, and, therefore, 3//3;y does not exist at the origin.

Supplementary Problems

In Problems 3.15 through 3.25, write the given differential equations in standard form.

2
3.15. xy' + y = 0 3.16. e"y'-x = y'
2
3.17. (y') 3 + y + y = sin x 3.18. xy' + cos(y' + y) = l
r
(y + y)
3.19. e ' = x 3.20. (y') 2 - 5/ + 6 = (x + y)(y - 2)
2
3.21. (x - y)dx + y dy = 0
2
3.24. (e * - y)dx + e*dy = 0
3.25. dy + dx = 0

In Problems 3.26 through 3.35, differential equations are given in both standard and differential form. Determine e whether
the equations in standard form are homogeneous and/or linear, and, if not linear, whether they are Bernoulli; determine
whether the equations in differential form, as given, are separable and/or exact.
3.26. y' = xy; xydx - dy = 0

3.27.

3.28.
3.29.