Page 175 - Schaum's Outline of Differential Equations

158 GRAPHICAL METHODS FOR SOLVING DIFFERENTIAL EQUATIONS [CHAP. 18

emanating from points on that isocline have the same slope, a slope equal to the constant that generated the
isocline. When they are simple to draw, isoclines yield many line elements at once which is useful for
constructing direction fields.

EULER'S METHOD
If an initial condition of the form

is also specified, then the only solution curve of Eq. (18.1) of interest is the one that passes through the initial
point (XQ, y 0).
To obtain a graphical approximation to the solution curve of Eqs. (18.1) and (18.2), begin by constructing
a line element at the initial point (x 0, y 0) and then continuing it for a short distance. Denote the terminal point
of this line element as (x 1, y^. Then construct a second line element at (x 1, y^ and continue it a short distance.
Denote the terminal point of this second line element as (x 2, y 2). Follow with a third line element constructed
at (x 2, y 2) and continue it a short distance. The process proceeds iteratively and concludes when enough of the
solution curve has been drawn to meet the needs of those concerned with the problem.
If the difference between successive x values are equal, that is, if for a specified constant h, h = x 1 - x 0
= x 2- x 1 = x 3- x 2 = ..., then the graphical method given above for a first-order initial-value problem is known
as Euler's method. It satisfies the formula

for n= 1,2,3, .... This formula is often written as

where
as required by Eq. (18.1).

STABILITY
The constant h in Eqs. (18.3) and (18.4) is called the step-size, and its value is arbitrary. In general, the smaller
the step-size, the more accurate the approximate solution becomes at the price of more work to obtain that solution.
Thus, the final choice of h may be a compromise between accuracy and effort. If h is chosen too large, then the
approximate solution may not resemble the real solution at all, a condition known as numerical instability. To
avoid numerical instability, Euler's method is repeated, each time with a step-size one half its previous value, until
two successive approximations are close enough to each other to satisfy the needs of the solver.

Solved Problems

18.1. Construct a direction field for the differential equation / = 2y - x.
U&Kf(x,y) = 2y-x.
I
At x= , y= !,/(!, 1) = 2(1) -1 = 1, equivalent to an angle of 45°.
Atx = l,y = 2,f(l, 2) = 2(2) -1=3, equivalent to an angle of 71.6°.
Atjc = 2, y= l,/(2, 1) = 2(1) -2 = 0, equivalent to an angle of 0°.
At x = 2, y = 2, f(2, 2) = 2(2) -2 = 2, equivalent to an angle of 63.4°.
Atjc= 1, y = -!,/(!, -1) = 2(-1)- l=-3, equivalent to an angle of-71.6°.
Atx = -2,y = -1, /(-2, -1) = 2(-l) - (-2) = 0, equivalent to an angle of 0°.