Hp 50g Graphing Calculator Manual do Utilizador descarregar pdf (Página 528)

100

101

Page 16-51

Except for a large peak at t = 0, the signal is mostly noise. A smaller vertical

scale (-0.5 to 0.5) shows the signal as follows:

Solution to specific second-order differential equations

In this section we present and solve specific types of ordinary differential

equations whose solutions are defined in terms of some classical functions, e.g.,

Bessel’s functions, Hermite polynomials, etc. Examples are presented in RPN

mode.

The Cauchy or Euler equation

An equation of the form x

⋅(d

y/dx

) + a⋅x⋅ (dy/dx) + b⋅y = 0, where a and b

are real constants, is known as the Cauchy or Euler equation. A solution to the

Cauchy equation can be found by assuming that y(x) = x

Type the equation as: ‘x^2*d1d1y(x)+a*x*d1y(x)+b*y(x)=0’ `

Then, type and substitute the suggested solution: ‘y(x) = x^n’ ` @SUBST

The result is: ‘x^2*(n*(x^(n-1-1)*(n-1)))+a*x*(n*x^(n-1))+b*x^n =0, which

simplifies to ‘n*(n-1)*x^n+a*n*x^n+b*x^n = 0’. Dividing by x^n, results in

an auxiliary algebraic equation: ‘n*(n-1)+a*n+b = 0’, or.

Θ If the equation has two different roots, say n

and n

, then the general

solution of this equation is y(x) = K

⋅x

+ K

⋅x

Θ If b = (1-a)

/4, then the equation has a double root n

= n

= n =

(1-a)/2, and the solution turns out to be y(x) = (K

+ K

⋅ln x)x

Legendre’s equation

An equation of the form (1-x

)⋅(d

y/dx

)-2⋅x⋅ (dy/dx)+n⋅ (n+1) ⋅y = 0, where n

is a real number, is known as the Legendre’s differential equation. Any solution

for this equation is known as a Legendre’s function. When n is a nonnegative

integer, the solutions are called Legendre’s polynomials. Legendre’s

polynomial of order n is given by

0)1(

=+⋅−+ bnan

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