Y 1, Y2 and Y3 are three solutions of the second-order differential equation, so Y2-Y 1 and Y3-Y 1 are two linearly independent solutions of the equation, so the general solution is: y = y1+c1(y2-y

The general solution of the equation is: y =1+c1(x-1)+C2 (x 2-1).

The second-order linear differential equation with constant coefficients is a differential equation in the form of y''+py'+qy=f(x), where p and q are real constants. When the free term f(x) is a continuous function defined on the interval I, that is, y''+py'+qy=0, it is called a second-order homogeneous linear differential equation with constant coefficients.

If the ratio of function y 1 to y2 is constant, it is said that y 1 is linearly related to y2; If the ratio of the function y 1 to y2 is not constant, it is said that y 1 is linearly independent of y2. The characteristic equation is λ 2+pλ+q = 0, and then the equation is solved according to the root of the characteristic equation.

Ordinary differential equations have a long history in higher mathematics. Because it is rooted in various practical problems, it continues to maintain the momentum of progress.

Second-order ordinary differential equations with constant coefficients play an important role in the theory of ordinary differential equations and are widely used in engineering technology, mechanics and physics. Commonly used solutions include undetermined coefficient method, polynomial method, constant variation method and differential operator method.