A second order differential equation is an equation involving the unknown function y, its derivatives y’ and y” and the variable x. in the mathematical form we can write d^2y/dx^2= f(x,y,y’). This equation is the basic form of second order differential equation.
Let’s discuss some types of second order differential, first is second order linear differential equation. The most general form of second order linear differential is p (t) y”+q(t)y’+r(t)y=g(t). Where p,q,r and g are continuous function, we can use this equation to study of the motion of spring. In the above equation if g(t) =0, the its is called homogenous linear equation so the form of second order linear homogeneous equation is p(t)y” + q(t)y’ + r(t)y = 0. As same if g(t) is not equal to 0 then the equation is called nonhomogeneous linear equation.
Now lets consider second order linear homogenous equation, now to solve this type of equation, there are two basic facts that we have to consider to solve linear homogeneous equation. The first fact says that if we know two solution y1(x) and y2(x) of such a equation, then the linear combination y= c1y1(x) +c2y2(x) is also a solution. The second facts says that if y1 and y2 are linearly independent solution of such equation (second order linear differential equation) and p(t) is never 0, then the general solution is given by y(x) = c1y1(x) + c2y2(x).
Generally this is not very easy to find the particular solution to second order linear differential equation, but it is always possible to do so if the coefficient p, q, and r are constant functions, that is, if the differential equation has the form
ay” + by’ + cy = 0, where a, b and c are constant and a is not equal to 0.
There is also some Second Order Differential Equation Solver, you can find it online. We can use MATLAB to solve second order differential, even we can find plot for the equation. In MATLAB we can solve second order ordinary differential equation with out any knowledge of numerical methods. MATLAB has number of tools to solve ordinary differential equation, but the two main tools are ode23 and ode45 which implement the version of RUNGA KUTTA 2nd/3rd and RUNGA KUTTA 4th/5th respectively. In MATLAB we can use LAPLAS transform, EULER’S method and Taylor methods. MATLAB has some additional solvers like ode113, ode23s, ode15s, ode23t and etc.
