# Solution To Specific Second-order Differential Equations; The Cauchy Or Euler Equation - HP F2226A - 48GII Graphic Calculator User Manual

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## 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
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.
If the equation has two different roots, say n
solution of this equation is y(x) = K
2
If b = (1-a)
/4, then the equation has a double root n
(1-a)/2, and the solution turns out to be y(x) = (K
2
⋅(d
2
2
y/dx
) + a⋅x⋅ (dy/dx) + b⋅y = 0, where a and b
2
n
(
a
) 1
n
b
0
and n
1
⋅x
n
⋅x
+ K
1
1
2
n
.
.
, then the general
2
n
.
2
= n
= n =
1
2
n
⋅ln x)x
+ K
.
1
2
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