HP 50g Graphing Calculator - Manual (Page 479)

Page 16-2
( H@)DISP) is not selected. Press ˜ to see the equation in the Equation
Writer.
An alternative notation for derivatives typed directly in the stack is to use ‘d1’ for
the derivative with respect to the first independent variable, ‘d2’ for the
derivative with respect to the second independent variable, etc. A second-
order derivative, e.g., d
2
x/dt
2
, where x = x(t), would be written as ‘d1d1x(t)’,
while (dx/dt)
2
would be written ‘d1x(t)^2’. Thus, the PDE ∂ 2
y/ ∂t 2
– g(x,y) ⋅
( ∂ 2
y/ ∂x 2
)
2
= r(x,y), would be written, using this notation, as ‘d2d2y(x,t)-
g(x,y)*d1d1y(x,t)^2=r(x,y)’.
The notation using ‘d’ and the order of the independent variable is the notation
preferred by the calculator when derivatives are involved in a calculation. For
example, using function DERIV, in ALG mode, as shown next
DERIV(‘x*f(x,t)+g(t,y) = h(x,y,t)’,t), produces the following expression:
‘x*d2f(x,t)+d1g(t,y)=d3h(x,y,t)’. Translated to paper, this
expression represents the partial differential equation x
⋅(∂f/∂t) + ∂g/∂t = ∂h/∂t.
Because the order of the variable t is different in f(x,t), g(t,y), and h(x,y,t),
derivatives with respect to t have different indices, i.e., d2f(x,t), d1g(t,y), and
d3h(x,y,t). All of them, however, represent derivatives with respect to the same
variable.
Expressions for derivatives using the order-of-variable index notation do not
translate into derivative notation in the equation writer, as you can check by
pressing
˜ while the last result is in stack level 1. However, the calculator
understands both notations and operates accordingly regarding of the notation
used.
Checking solutions in the calculator
To check if a function satisfy a certain equation using the calculator, use
function SUBST (see Chapter 5) to replace the solution in the form ‘y = f(x)’ or ‘y
= f(x,t)’, etc., into the differential equation. You may need to simplify the 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)))+ ω0^2*u(t) = 0’,‘u(t)=A*SIN (ω0*t)’) `