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.