P 4 0.4 0.064 (0.464)(0.1) = 0.046
P 5 0.5 0.110
A Caution: Euler’s method approximates the solution by substituting short line segments in place of
the actual curve. It can be quite accurate when the step sizes are small, but only if the curve does not
have discontinuities, cusps, or asymptotes.
For example, the reader may verify that the curve for the domain solves the
differential equation with initial condition y = −1 when x = 2. The domain restriction is
important. Recall that a particular solution must be differentiable on an interval containing the initial
point. If we attempt to approximate this solution using Euler’s method with step size Δx = 1, the first
step carries us to point (3, −3), beyond the discontinuity at and thus outside the domain of the
solution. The accompanying graph (Figure N9–7) shows that this is nowhere near the solution curve
with initial point y = 1 when x = 3 (and whose domain is ). Here, Euler’s method fails because it
leaps blindly across the vertical asymptote at
Always pay attention to the domain of any particular solution.
BC ONLY
FIGURE N9–7D. SOLVING FIRST-ORDER DIFFERENTIAL EQUATIONS
ANALYTICALLY
In the preceding sections we solved differential equations graphically, using slope fields, and
numerically, using Euler’s method. Both methods yield approximations. In this section we review
how to solve some differential equations exactly.
Separating Variables