This equation is a second-order linear homogeneous differential equation, where y is a function of x and w is a constant. It can be solved to find the function y(x) that satisfies the equation.
1. Identify the coefficients of y''(x), y'(x), and y(x). In this case, they are 1, 2/x, and w^2 respectively.
2. Rewrite the equation in the standard form y''(x)+p(x)y'(x)+q(x)y(x)=0, where p(x) and q(x) are the coefficients.
3. Find the roots of the characteristic equation r^2+p(x)r+q(x)=0. In this case, the roots are -w and -w^2.
4. Write the general solution as y(x)=c1e^(-wx)+c2e^(-w^2x), where c1 and c2 are constants.
5. Use initial conditions or boundary conditions to solve for the constants and find the specific solution.
The constant w determines the behavior of the solution. It can affect the frequency, amplitude, and stability of the solution. A change in w can result in different types of solutions, such as oscillatory or non-oscillatory.
This equation can be solved both analytically and numerically. Analytical solutions involve finding a closed-form expression for the function y(x), while numerical solutions involve using algorithms to approximate the solution.
This type of equation is commonly used in physics and engineering to model various systems such as oscillating springs, electrical circuits, and vibrating structures. It can also be used in economics and finance to model population growth and stock prices.