Ordinary Differential Equations

Ordinary Differential Equations

1. is single variable DE - depens on only x

2. Separable DE

   • y' = g(x)/f(y) y' f(y) = g(x)

3. Linear ODE

   • First Order Linear n=1
     • y' = a(x)*y + bx
       • Case 1: a=const b=0
         • y' = a*y
           • Solution: y = k*exp(ax)
       • Case 2: a=function b=0
         • y' = a(x) * y
           • Solution: y(x)=kexp(A(x))
       • General solution = Homogenous + Particular
         • Particular
           • y' = 0
           • Not always obvious how to find P.S., so we use:
             • Right-Hand Side Method
               • 1. Homogenous solution (second member=0), y(x)=k*exp(ax)
               • 2. Particular solution can be of the forms:
                 • Second degree polynomial ax^2+bx+c
                 • Trigonometric form a*cos(x)+ b*sin(x)
                 • Product of exponential function and a first order polynomial Q(x)*exp(x), where Q(x)=ax^2+bx+c
                 • Sum of a first order polynomial, exp and trig. functions ax+b+m*exp(x)+Alpha*cos(x)+Beta*sin(x)
             • The Variation of Coefficents Method
               • Homogenous Solution: y = k*exp(A(x)), where A'(x)=a(x)
               • Particular Solution: y = k(x)*exp(A(x)), where we need to find k(x) firstly to solve
             • Substitution Technique
               • Bernoulli Equation y'+P(x)*y=Q(x)*y^n
                 • Substitute u=y^(1-n) into Bernoulli Equation
               • y = e^u
               • u = y'
         • Homogenous
           • Second member = 0
   • Homogenous = without second member
     • y'+5xy=0
   • Constant - coefficent
     • 2y''-3y'+5y=0
   • To solve Linear ODE, General Solution = Particular + Homogenous:
     • Particular Solution
       • Integration of both sides
     • Homogenous soolution
       • Second member = 0