Second Order Differential Equation

Typical form of second order differential equation

                 y"=f(x,y,y')

Solutions

1. y' and y missing :   y"=f(x) integrate twice with respect to x
2. y missing : y"=f(y',x) 
1. put p=y'    =>  p'=y"=f(p,x) this become first order equation
3. y' and x missing : y"=f(y)     => put p=y'   =>    => this is seprable with respect to p and y.
4. x missing   : y"=f(y',y) => put p=y' =>   

Second order linear homogeneous  equation with constant coefficients 

1. form(characteristic) : y"+a₁y'+a₀y=0   => λ²+a₁λ+a₀=0  
1. Solving this quadratic equation got 3 kind of roots.
1. real and distinct λ1,λ2 then solution
1. y=c₁e^(λ1x)+c₂e^(λ2x)
2. λ₁=a+ib and λ₂=a-ib complex conjugates
1. y=c₁e^ax cos(bx)+c₂e^ax sin(bx)
3. λ1=λ2
1. y=c₁e^λ1x+c₂x^λ1x
2. equation should be linear,  homogeneous  and constant coefficient  only.
3. This solution also workout for similar n^th order equations.

Non-Homogeneous Equations

        L(y)=ϕ(x)  then General solution GS  y=y_h+y_p  =>   y=y_c+y_p

there is three way to find y_p

1. "Guesswork"
2. Annihilator
3. D-operator

Guesswork Method

1. Polynomial function y_p=g(x)=p₀+p₁x+p₂x²+........+p_nxⁿ  Solve this equation with differentiating wrt x and equates coefficients of xⁿ
2. exponential function  y_p=g(x)=Ae^ax      Solve this equation with differentiating wrt x
3.  Trigonometrical function    y_p=g(x)=A sin βx + B cos βx    Solve this equation with differentiating wrt x
4. Above 3 will work for nth order differential equation.

Eluer equation : linear ODE with variable coefficients

            use y=x^λ