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result by using function EVAL to verify the solution. For example, to check that
u = A sin
ω
o
t is a solution of the equation d
2
u/dt
2
+
ω
o
2
⋅
u = 0, use the
following:
In ALG mode:
SUBST(‘
∂
t(
∂
t(u(t)))+
ω
0^2*u(t) = 0’,‘u(t)=A*SIN (
ω
0*t)’)
`
EVAL(ANS(1))
`
In RPN mode:
‘
∂
t(
∂
t(u(t)))+
ω
0^2*u(t) = 0’
`
‘u(t)=A*SIN (
ω
0*t)’
`
SUBST EVAL
The result is ‘0=0’.
For this example, you could also use: ‘
∂
t(
∂
t(u(t))))+
ω
0^2*u(t) = 0’ to enter the
differential equation.
Slope field visualization of solutions
Slope field plots, introduced in Chapter 12, are used to visualize the solutions
to a differential equation of the form dy/dx = f(x,y). A slope field plot shows
a number of segments tangential to the solution curves, y = f(x). The slope of
the segments at any point (x,y) is given by dy/dx = f(x,y), evaluated at any
point (x,y), represents the slope of the tangent line at point (x,y).
Example 1 -- Trace the solution to the differential equation y’ = f(x,y) = sin x
cos y, using a slope field plot. To solve this problem, follow the instructions in
Chapter 12 for
slopefield
plots.
If you could reproduce the slope field plot in paper, you can trace by hand
lines that are tangent to the line segments shown in the plot. This lines
constitute lines of y(x,y) = constant, for the solution of y’ = f(x,y). Thus, slope
fields are useful tools for visualizing particularly difficult equations to solve.
In summary, slope fields are graphical aids to sketch the curves y = g(x) that
correspond to solutions of the differential equation dy/dx = f(x,y).
