Page 46 - Schaum's Outline of Differential Equations

CHAP. 4] SEPARABLE FIRST-ORDER DIFFERENTIAL EQUATIONS 29

Integrating both sides of this last equation with respect to x, we obtain

In the second integral, make the change of variables y = y(x), hence dy = y'(x) dx. The result of this substitution is (4.2).

4.19. Prove that every solution of system (4.3) is a solution of (4.4).
Following the same reasoning as in Problem 4.18, except now integrating from x = X Q to x = x, we obtain

The substitution y = y(x) again gives the desired result. Note that as x varies from X Q to x, y will vary from y(x Q) = y Q
to y(x) = y.

4.20. Prove that if / =f(x, y) is homogeneous, then the differential equation can be rewritten as / = g(ylx),
where g(ylx) depends only on the quotient ylx.
We have that/(jc, y) =f(tx, ty). Since this equation is valid for all t, it must be true, in particular, for t = 1/x.
Thus, f(x, y) =/(!, ylx). If we now define g(ylx) =/(!, ylx), we then have y' =f(x, y) =/(!, ylx) = g(ylx) as
required.
Note that this form suggests the substitution v = ylx which is equivalent to (4.6). If, in the above, we had set
t= lly, then/(X y) =f(xly, 1) = h(xly), which suggests the alternate substitution (4.9).

4.21. A function g(x, y) is homogeneous of degree n if g(tx, ty) = fg(x, y) for all t. Determine whether the
following functions are homogeneous, and, if so, find their degree:
2
2
3
x/y
2
(a) xy + y , (b) x + y sin (ylx) , (c) x + xy e, , and (d) x + xy.
2
2
(a) (tx)(ty) + (ty) 2 = t (xy + y ); homogeneous of degree two.
homogeneous of degree one.
2 Klty
2 xly
(c) (txf + (tx)(ty) e = ^(x 3 + xy e ); homogeneous of degree three.
2
(d) tx + (tx)(ty) = tx+ t xy; not homogeneous.
4.22. An alternate definition of a homogeneous differential equation is as follows: A differential equation
M(x, y) dx + N(x, y) dy = 0 is homogenous if both M(x, y) and N(x, y) are homogeneous of the same
degree (see Problem 4.21). Show that this definition implies the definition given in Chapter 3.

If M(x, y) and N(x, y) are homogeneous of degree n, then

Supplementary Problems

In Problems 4.23 through 4.45, solve the given differential equations or initial-value problems.

3
4.23. x dx + y dy = 0 4.24. x dx - y dy = 0