91Ó°ÊÓ

The quadratic formula is a fundamental tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). Here, \( a, b, \) and \( c \) are constants.
The formula used to find the values of \( x \) that satisfy the equation is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides two solutions due to the \( \pm \) symbol, representing the two possible roots of the equation.
In our example problem—solving a quadratic equation for \( m \) derived from a Cauchy-Euler differential equation—the quadratic formula helps find the values of \( m \) that form the basis of our solution.
Working through this, you begin with the equation \( m^2 + 2m - 1 = 0 \), where \( a = 1 \), \( b = 2 \), and \( c = -1 \).
The discriminant, \( b^2 - 4ac \), is calculated as \( 8 \), indicating two distinct real solutions. Substituting these into the quadratic formula gives the roots \( m = -1 \pm \sqrt{2} \).
These roots are crucial for constructing the general solution of the differential equation.

The general solution of a differential equation represents a family of possible solutions that include arbitrary constants until more specific conditions are given, such as initial or boundary conditions. For Cauchy-Euler differential equations, these solutions can often be expressed in a general form.
Returning to our differential equation problem, the roots found using the quadratic formula, \( m = -1 + \sqrt{2} \) and \( m = -1 - \sqrt{2} \), lead directly to the general solution.
This solution is a linear combination of two functions, each corresponding to one of the roots:

• \( y = C_1 x^{-1 + \sqrt{2}} + C_2 x^{-1 - \sqrt{2}} \)

Here, \( C_1 \) and \( C_2 \) are arbitrary constants that will be determined by any additional conditions provided beyond the current scope (\( x > 0 \) in this instance).
The exponents in \( x \) form suggest these solutions will vary with \( x \) in a non-integer fashion, reflecting the nature of the roots found.

When solving Cauchy-Euler differential equations, a common technique begins with assuming a specific solution form. This strategy recognizes the characteristic nature of these equations.
In our exercise, this approach means assuming that the solution can be written as \( y = x^m \), where \( m \) is a constant to be determined.
This assumption is particularly effective because the equation involves terms that are powers of \( x \).
By assuming \( y = x^m \), we can simplify the problem by reducing it to finding the appropriate \( m \) that satisfies the equation when substituted back into the differential equation.
The derivatives, \( y' = m x^{m-1} \) and \( y'' = m(m-1) x^{m-2} \), are utilized to simplify the differential equation:

• \( m(m-1)x^{m} + 3mx^m - x^m = 0 \)

The solving process confirms which values of \( m \) satisfy this equation, unlocking the structure of the solution.