Second – Order Differential Equations and Solving Using MATLAB

This study introduced real life application of second order differential equation. We basically discussed about different types of differential equation and the solution of second order differential equation and application of second order differential equation in different field of science and technology using MATLAB. The problem of existence of periodic solution is studied for the second order delay differential equation with a singularity of repulsive type:

x''(t) + f(x(t)x'(t) + φ(t)x(t - τ₁ ) - g(x(t - τ₂ )) = h(t).

Where τ₁ and τ₂ are constants, g(x) is singular at x = 0, φ and h are T − periodic functions. By using a continuation theorem of coincidence degree theory, a new result on the existence of positive periodic solutions is obtained. The interesting is that the sign of function φ(t) is allowed to change for t ∈ [0, T]. The study followed applied analytical approach and reached the most important results.

The system is modelled by the second – order linear non-homogeneous differential equation y"+ 5 y' + 6y = 10Sin (ωt) with initial conditions y(0)=0 and y'(0)=5. The general solution consists of two distinct components  y(t) = y_h(t) + y_p(t).

ω = 2 rad/s substituting ω = 2 into the equations gives exact values for the coefficients: A= - 0.9615, B= 0.1923, C₁=1.9231, C₂=-1.9231, The final time-domain solution is: y(t) = 1.9231e-^2t - 1.9231e-^3t -0.961.

Initial state (t = 0): The displacement starts at   while the initial velocity y’(0)=5 causes an immediate positive movement.

Damping Phase ( t < 2s ): The system’s natural damping (damping factor = 5) quickly suppresses the exponential terms.

Long – term state ( t > 3s ): The system synchronizes entirely with the external driving force, oscillating strictly as a sine wave with amplitude R= √(A²+B² ) ≈ 0.98 and frequency ω = 2 rad/s.

We recommend that researchers conduct further research on solutions to these differential equations using MATLAB and other software to solve and plot them.