Page 35 - Schaum's Outline of Differential Equations

18 CLASSIFICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 3

(a) The equation is homogeneous, since

(b) The equation is not homogeneous, since

(c) The equation is homogeneous, since

(d) The equation is not homogeneous, since

3.10. Determine if the following differential equations are separable:
2
z
2 2
(a) sinxdx + y dy = 0 (b) xy dx - x y dy = 0 (c) (1 + xy)dx + y dy = 0
(a) The differential equation is separable; here M(x, y) = A(x) = sin x and N(x, y) = B(y) = y . 2
(b) The equation is not separable in its present form, since M(x, y) = xy 2 is not a function of x alone. But if we
2 2
divide both sides of the equation by x y , we obtain the equation (llx)dx+ (-l)dy = 0, which is separable.
Here, A(x) = llx and B(y) = -1.
(c) The equation is not separable, since M(x, y) = 1 + xy, which is not a function of x alone.
3.11. Determine whether the following differential equations are exact:
3
2
2
(a) 3x ydx+(y + x )dy = 0 (b) xydx + y dy = 0
3
2
2
(a) The equation is exact; here M(x, y) = 3x y, N(x, y) = y + x , and 3M3;y = 3AV3;c = 3x .
2
(b) The equation is not exact. Here M(x, y) = xy and N(x, y) = y ; hence 3M3;y = x, 3AV3;c = 0, and 3M3;y ^ 3AV3;c.
3.12. Determine whether the differential equation / = ylx is exact.
Exactness is only defined for equations in differential form, not standard form. The given differential equation
has manv differential forms. One such form is given in Problem 3.5. Eci. (1). as

Here

and the equation is not exact. A second differential form for the same differential equation is given in Eq. (3) of
Problem 3.5 as