General Solution of Second Order Differential Equation

Find the general solution and a second solution of the differential equation $x^2 y'' + 5x y' - 5y = 0$ given that $y_1 = x$ is a known solution.

In this problem, you are tasked with solving a second-order homogeneous differential equation with variable coefficients where one solution is already given. The equation is of the form x squared times the second derivative of y plus five times x times the first derivative of y minus five times y equals zero. You are provided with a known solution $y_1 = x$, and are asked to find the general solution as well as a second linearly independent solution. This problem is common in differential equations courses, especially when learning about reduction of order as a technique to find a second solution when one solution is already known.

The concept of finding the general solution involves understanding the properties of linear differential equations and exploiting the Wronskian to test for linear independence of solutions. In this case, given one solution, you can use the method of reduction of order to find another function that satisfies the differential equation. This technique involves assuming a form for the second solution, typically as a product of the known solution and an unknown function, and then deriving an equation for this unknown function.

Besides reinforcing techniques specific to differential equations, this problem also deepens your understanding of the structure and solution behavior of second-order differential equations. Specifically, it illustrates how solutions can be deduced systematically rather than guessed. It is advantageous to develop a solid foundation in these methods as they are applicable to a wide range of problems in engineering, physics, and applied mathematics, where such equations frequently occur.