Solving the Cauchy-Euler Equation x2y'' - xy' - y 0

The Cauchy-Euler equation is a type of second-order linear differential equation with variable coefficients. The given equation is:

x2y'' - xy' - y 0

Transformation of Variables

To simplify the equation, we can use the substitution: x et. This transformation is common because it converts the Cauchy-Euler equation into a linear differential equation with constant coefficients.

Substitution:

Let y xlambda;. Then, the first and second derivatives are:

Substituting into the Equation:

Substitute these derivatives into the original differential equation:

x2lambda;(lambda;-1)xlambda;-2 - xlambda;xlambda;-1 - xlambda; 0

After simplifying, we get:

lambda;(lambda;-1) - lambda; - 1 0

Further simplification gives:

lambda;2 - lambda; - 1 0

Solving this quadratic equation for lambda; yields:

lambda;2 -1

lambda; ±i

General Solution

For complex roots of the form lambda; alpha; ±beta;i, the general solution to the Cauchy-Euler equation is:

y C1xalpha;cos(beta;lnx) C2xalpha;sin(beta;lnx)

In this case, since alpha; 0, the general solution simplifies to:

y C1cos(lnx) C2sin(lnx)

Where C1 and C2 are arbitrary constants.

Graphical Representation

To visualize the solution, we can plot the general solution:

y C1cos(lnx) C2sin(lnx)

For selected values of C1 and C2, the resulting plots show the behavior of the solutions in the x-y plane.