HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 481

result by using function EVAL to verify the solution. For example, to check that u = A sin ωot is a solution of the equation d2u/dt2 + ωo2⋅u = 0, use the following: In ALG mode: SUBST('∂t(∂t(u(t)))+ ω0^2*u(t) = 0','u(t)=A*SIN (ω0*t)') ` EVAL(ANS(1)) ` In RPN mode: '∂t(∂t(u(t)))+ ω0^2*u(t) = 0' ` 'u(t)=A*SIN (ω0*t)' ` SUBST EVAL The result is '0=0'. For this example, you could also use: '∂t(∂t(u(t))))+ ω0^2*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). Page 16-3