SECOND ORDER DiIFFERENTIAL EQUATIONS

Second order differential equation is an equation that relates a function, its first derivative, and its second derivative.

If p(x) and q(x) are constants than the equation is called second order linear differential equation with constant coefficients.

Second order differential equation can be classified into 2 types:

1.Homogeneous differential equation: if Q=0 in the above equation then the differential equation is said to be Homogeneous differential equation.

The solution of the homogeneous differential equation of 2nd order can be classified based on the nature of the roots of an auxillary equation:

1.Roots are real and unequal

2.Roots are imaginary

Where the roots r1 and r2 are imaginary which means r1= alpha +i*betta and r2= alpha-i*betta

3.Roots are real and equal

Where the roots r1 is equal to r2.

2.Non-Homogeneous differential equation: is the solution is given as the sum of complementary function and particular integral that is the C.F of the solution of homogeneous differential equation and PI is obtained based on the nature of Q.

NATURE OF Q(x);

If Q=e^ax, then PI= Q/f(D) =e^ax/f(a)
If Q=sinax then PI= Q/f(D) =sinax/f(D)= sinax/replace D^2 by -a^2
If Q=x^m then PI= Q/f(D)=x^m/1+f(D)= [1+f(D)]^-1*x^m
If Q=e^ax*v then PI=Q/f(D)=e^ax*v/f(D)=e^ax 1/f(D+a)*v
If Q=xv where v=f(x) then PI=Q/f(D)= xv/f(D)=[x*1/f(D)*v-f'(D)/[f(D)]^2*v

When the acceleration of the particle is proportional to its displacement from a fixed point and is always directed towards it then the motion is called simple harmonic motion.

VELOCITY FORMULA:

DISPLACEMENT FORMULA :

TIME PERIOD FORMULA

The vibrating stings means that the frequency of a stretched string's vibration and its resonating length fluctuates inversely.

This law states that the force required to manitaining string stretch x units beyond its natural length is proportional to x that is fv=kx.

Where k is positive constant called spring constant.

If the spring is stretched or compressed x units from its natural length than it exerts a force that is proportional to x that is restoring force=-kx.
If we ignore resisting forces than the Newton's 2nd law is mF=ma

Damped vibrations are characterized by a decrease in amplitude over time due to energy dissipation often caused by friction.

The solution to this equation describes how the amplitude decays and how the frequency of oscillation is affected by damping.

CASE 1: OVER DAMPING

c^2-4mk>0

CASE 2: CRITICAL DAMPING

c^2-4mk>0

CASE 3; UNDER DAMPING

c^2-4mk<0