An examination of the forces on a spring-mass system results in a differential equation of the form \[mx″+bx′+kx=f(t), \nonumber\] where mm represents the mass, bb is the coefficient of the damping force, \(k\) is the spring constant, and \(f(t)\) represents any net external forces on the system. Our job is to show that the solution is correct. Second Order Linear Homogeneous Differential Equations with Constant Coefficients. Substituting \(y(x)\) into the differential equation, we have Find the general solution of the given second-order differential equation. Let the general solution of a second order homogeneous differential equation be If the general solution \({y_0}\) of the associated homogeneous equation is known, then the general solution for the nonhomogeneous equation can be found by using the method of variation of constants. To prove \(y(x)\) is the general solution, we must first show that it solves the differential equation and, second, that any solution to the differential equation can be written in that form. Consider a differential equation of type \[{y^{\prime\prime} + py’ + qy }={ 0,}\] where \(p, q\) are some constant coefficients. Theroem: The general solution of the second order nonhomogeneous linear equation y″ + p(t) y′ + q(t) y = g(t) can be expressed in the form y = y c + Y where Y is any specific function that satisfies the nonhomogeneous equation, and y c = C 1 y 1 + C 2 y 2 is a general solution of the corresponding homogeneous equation y″ + p(t) y′ + q(t) y = 0. The first thing we want to learn about second-order homogeneous differential equations is how to find their general solutions. We do this by substituting the answer into the original 2nd order differential equation. A General Solution of an nth order differential equation is one that involves n necessary arbitrary constants. We have a second order differential equation and we have been given the general solution. Read It Watch It Talk to a Tutor We need to find the second derivative of y: y = c 1 sin 2x + 3 cos 2x. For each equation, numerical examples are presented to illustrate the proposed approach. 2y" - 5y' + 6y = 0 y(x) = Need Help? First derivative: `(dy)/(dx)=2c_1 cos 2x-6 sin 2x` General Solution of a Differential Equation. The formula we’ll use for the general solution will depend on the kinds of roots we find for the differential equation. Second-order constant-coefficient differential equations can be used to model spring-mass systems. General solution to fractional differential equations are detected, based on conformable fractional derivative. If we solve a first order differential equation by variables separable method, we necessarily have to introduce an arbitrary constant as soon as the integration is performed.