91Ó°ÊÓ

The given differential equation, \(2 x^{2} y^{\beta \beta} - 2 x y^{\beta} + 20 y = 0\), is classified as a second-order linear homogeneous differential equation with variable coefficients. This means that the coefficients (functions multiplying the derivatives and the function itself) depend on the variable \(x\). In this case:

• The coefficient of \(y^{\beta \beta}\) is \(2x^2\).
• The coefficient of \(y^{\beta}\) is \(-2x\).
• The coefficient of \(y\) is \(20\).

Having variable coefficients generally complicates the process of solving these types of equations because techniques for solving them vary compared to equations with constant coefficients. To handle equations with variable coefficients, solutions often involve methods like transformation into simpler forms or the use of specialized functions.

To solve the differential equation, we form what is known as the auxiliary equation or characteristic equation. This involves assuming a solution of the form \(y = x^r\). By substituting this assumed solution and its derivatives back into the original differential equation, we create an algebraic equation for \(r\), called the auxiliary equation.
For our specific problem:

• The first derivative is \(y^{\beta} = r x^{r-1}\).
• The second derivative is \(y^{\beta \beta} = r(r-1) x^{r-2}\).

Substituting these into the differential equation, we get: \[x^{2} r (r-1) x^{r-2} - x r x^{r-1} + 10 x^r = 0 \]. Simplifying this, we find: \[r(r-1) - r + 10 = 0 \], which reduces to the quadratic auxiliary equation: \[r^2 - 2r + 10 = 0 \]. Solving this quadratic equation using the quadratic formula, we get the roots: \[r = 1 \pm 3i \]. These roots help us form the general solution to the differential equation.

With the roots of the auxiliary equation, we can now form the general solution of the differential equation. Since the roots \(r = 1 \pm 3i \) are complex, the general solution is composed of both the real and imaginary parts:

• The general form of the solution for complex roots \(r = \alpha \pm \beta i\) is \(y = x^{\alpha} (C_1 \cos(\beta \ln x) + C_2 \sin(\beta \ln x))\).

For our problem: \[y(x) = x (C_1 \cos(3 \ln x) + C_2 \sin(3 \ln x)) \]. This form is obtained by:

• Combining Euler's formula for complex exponentials.
• Ensuring the solution satisfies the initial differential equation.

The constants \(C_1\) and \(C_2\) are determined by initial conditions or specific boundary values provided in the problem. This is the finalized form of the general solution for the given second-order linear homogeneous differential equation with variable coefficients.